Chapter 6 — Front Matter
Purpose and Overview
Chapter 5 introduced the concept of the inverse matrix and connected it to the idea of reversibility. When a matrix has an inverse, the transformation it represents can be undone, and the system \(A\mathbf{x} = \mathbf{b}\) has a unique solution for every right-hand side. But that chapter left an important question unanswered: how do you know, quickly and precisely, whether a matrix has an inverse at all? Is there a number — a single calculated quantity — that captures whether a matrix collapses space or preserves it?
This chapter answers that question. The determinant is that number. Associated with every square matrix, the determinant is a scalar quantity that encodes geometric information about the transformation the matrix represents. A nonzero determinant signals that the transformation stretches, compresses, or reflects space — but does not collapse it. A zero determinant signals that the transformation crushes space into a lower dimension, making the matrix singular.
This chapter develops the determinant from the ground up. You will learn what the determinant measures geometrically, how to compute it for \(2 \times 2\) and \(3 \times 3\) matrices, how its properties simplify calculation, how it connects to row operations, and how it relates to invertibility, linear dependence, and systems. You will also see Cramer's Rule as an elegant theoretical application, and you will explore how determinants appear in geometry, physics, economics, and computer graphics.
Determinants are more than a formula. They are a measurement of how a matrix interacts with space.
Opening Question
Suppose you are designing a computer graphics engine. Your engine uses matrices to transform objects: stretching a logo, rotating a shape, flipping an image, shearing a polygon. Most transformations preserve the recognizable structure of an object. But what happens if you accidentally apply a transformation that flattens a two-dimensional figure into a single line? The object would vanish from the screen. How can you detect — without visually inspecting every object — whether a transformation matrix will collapse your shapes into a lower dimension?
The determinant of the matrix gives you the answer instantly. A determinant of zero means collapse. A nonzero determinant tells you how much the transformation scales areas or volumes, and even whether it flips orientation. This is the practical power of the determinant: a single number that tells you everything about the scaling behavior of a matrix.
14Sections
11Objectives
14Key Terms
12+Worked Examples
Learning Objectives
- 1Explain what the determinant of a matrix measures geometrically.
- 2Compute the determinant of any \(2 \times 2\) matrix using the standard formula.
- 3Interpret the determinant as an area scaling factor and identify orientation.
- 4Compute the determinant of a \(3 \times 3\) matrix using cofactor expansion along any row or column.
- 5Interpret the \(3 \times 3\) determinant as a volume scaling factor.
- 6State and apply the major properties of determinants, including the effects of row operations.
- 7Use determinant properties to simplify computation.
- 8Connect determinants to invertibility and linear dependence.
- 9Apply Cramer's Rule to solve small linear systems.
- 10Recognize the real-world contexts in which determinants arise.
- 11Identify common determinant errors and avoid them.
Sections 6.1 – 6.3
Geometric Meaning and the \(2 \times 2\) Determinant
What the Determinant Measures · The \(2 \times 2\) Formula · Area and Orientation
Section 6.1
What a Determinant Measures
Before computing anything, let us build the right mental model. A determinant is a number associated with a square matrix that answers a specific question about geometry. When you apply a matrix to a region of space, the region changes. Points move, distances stretch or shrink, angles shift. But the most fundamental geometric quantity that a linear transformation affects is area (in two dimensions) and volume (in three dimensions). The determinant measures exactly how much that area or volume changes.
More precisely, if you take any shape in \(\mathbb{R}^2\) and apply the transformation represented by a \(2 \times 2\) matrix \(A\), the area of the resulting shape equals \(|\det(A)|\) times the original area. If you take any solid in \(\mathbb{R}^3\) and apply a \(3 \times 3\) matrix, the volume of the resulting solid equals \(|\det(A)|\) times the original volume. This is called the scaling factor of the transformation.
The sign of the determinant carries additional geometric meaning. A positive determinant means the transformation preserves orientation — the sense of "clockwise" and "counterclockwise" stays the same. A negative determinant means the transformation reverses orientation — it includes a reflection. The magnitude of the determinant gives the scale factor; the sign gives the orientation.
The most important special case is \(\det(A) = 0\). When the determinant is zero, the transformation collapses space: it maps a two-dimensional region onto a line, or maps a three-dimensional solid onto a plane or a line. Some dimension is lost. The transformation cannot be reversed, so the matrix is not invertible. This connection between zero determinant and non-invertibility is one of the most important facts in all of linear algebra.
| Value of \(\det(A)\) | Geometric Meaning |
|---|
| \(\det(A) > 0\) | Preserves orientation; scales area or volume by \(\det(A)\) |
| \(\det(A) < 0\) | Reverses orientation; scales area or volume by \(|\det(A)|\) |
| \(\det(A) = 0\) | Collapses dimension; transformation is not invertible; matrix is singular |
Diagram — The Unit Square Under Transformation
Imagine the unit square in \(\mathbb{R}^2\) with corners at \((0,0)\), \((1,0)\), \((1,1)\), and \((0,1)\). When you apply matrix \(A\) to this square, the corners move to new positions and the resulting figure is a parallelogram. The area of that parallelogram equals \(|\det(A)|\). When \(\det(A) > 0\), the vertices are traversed counterclockwise as in the original — orientation is preserved. When \(\det(A) < 0\), the vertices are traversed clockwise — the orientation is flipped.
Section 6.2
Determinants of \(2 \times 2\) Matrices
The simplest determinant to compute is that of a \(2 \times 2\) matrix. For
\[ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \]
the determinant of \(A\), written \(\det(A)\) or \(|A|\), is defined as:
The \(2 \times 2\) Determinant Formula
\[ \det(A) = ad - bc \]
Multiply the entries along the main diagonal (\(a\) and \(d\)), then subtract the product of the entries along the other diagonal (\(b\) and \(c\)).
Notation Note
The determinant is also written using vertical bars: \(\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc\). Be careful not to confuse these vertical bars with absolute value notation. Vertical bars around a matrix specifically denote the determinant.
Example 6.2.1
Computing \(2 \times 2\) determinants and interpreting results
- ProblemCompute the determinant of each matrix:
\[A = \begin{pmatrix} 3 & 5 \\ 1 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 2 & -3 \\ -4 & 6 \end{pmatrix}, \quad C = \begin{pmatrix} -1 & 2 \\ 3 & -6 \end{pmatrix}.\]
- ThinkApply the formula \(\det = ad - bc\) to each matrix. Then interpret the result geometrically.
- Calculate
\[\det(A) = (3)(4) - (5)(1) = 12 - 5 = 7\]
\[\det(B) = (2)(6) - (-3)(-4) = 12 - 12 = 0\]
\[\det(C) = (-1)(-6) - (2)(3) = 6 - 6 = 0\]
- CheckVerify each product carefully. Notice that in matrix \(B\), row 2 is \(-2\) times row 1, and in matrix \(C\), row 2 is \(-3\) times row 1 — proportional rows always produce a zero determinant.
- InterpretMatrix \(A\): \(\det(A) = 7\). The transformation scales areas by 7 and preserves orientation. Matrix \(A\) is invertible. Matrices \(B\) and \(C\): both have \(\det = 0\). Neither is invertible. Each collapses the plane onto a line.
Answers: det(A) = 7 (invertible); det(B) = 0 (singular); det(C) = 0 (singular)
Example 6.2.2
A determinant with fractions and negatives
- ProblemCompute \(\det(A)\) for \(A = \begin{pmatrix} -4 & \frac{1}{2} \\ 6 & -3 \end{pmatrix}\).
- Calculate
\[\det(A) = (-4)(-3) - \left(\tfrac{1}{2}\right)(6) = 12 - 3 = 9\]
- CheckNegative times negative is positive; positive times positive is positive. Both products are computed with correct sign rules.
- InterpretThis transformation scales areas by a factor of 9 and preserves orientation. The matrix is invertible.
Answer: det(A) = 9
Section 6.3
Area Scale and Orientation
Now let us see concretely why the formula \(ad - bc\) measures area. Consider the two column vectors of \(A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\):
\[\mathbf{v}_1 = \begin{pmatrix} a \\ c \end{pmatrix}, \quad \mathbf{v}_2 = \begin{pmatrix} b \\ d \end{pmatrix}.\]
These two vectors, drawn from the origin, define a parallelogram. The area of that parallelogram is exactly \(|ad - bc| = |\det(A)|\). Start with the unit square — the square formed by the standard basis vectors \(\mathbf{e}_1\) and \(\mathbf{e}_2\) — which has area 1. When you apply matrix \(A\) to the unit square, \(\mathbf{e}_1\) maps to \(\mathbf{v}_1\) and \(\mathbf{e}_2\) maps to \(\mathbf{v}_2\). The unit square transforms into the parallelogram spanned by \(\mathbf{v}_1\) and \(\mathbf{v}_2\), whose area is \(|\det(A)|\). The sign of \(\det(A)\) captures whether the transformation flips the plane.
Example 6.3.1
Area of a parallelogram defined by two vectors
- ProblemVectors \(\mathbf{v}_1 = \begin{pmatrix} 3 \\ 1 \end{pmatrix}\) and \(\mathbf{v}_2 = \begin{pmatrix} -1 \\ 4 \end{pmatrix}\) are adjacent sides of a parallelogram based at the origin. Find the area.
- RepresentForm the matrix whose columns are \(\mathbf{v}_1\) and \(\mathbf{v}_2\):
\[A = \begin{pmatrix} 3 & -1 \\ 1 & 4 \end{pmatrix}.\]
- Calculate
\[\det(A) = (3)(4) - (-1)(1) = 12 + 1 = 13\]
- CheckThe area is \(|\det(A)| = 13\). Since the determinant is positive, \(\mathbf{v}_2\) is counterclockwise from \(\mathbf{v}_1\).
- InterpretThe parallelogram with sides \(\mathbf{v}_1\) and \(\mathbf{v}_2\) has area 13 square units. Orientation is preserved.
Answer: Area = 13 square units
Diagram — Parallelogram Area
Draw the coordinate plane. Place \(\mathbf{v}_1 = (3,1)\) and \(\mathbf{v}_2 = (-1, 4)\) as arrows from the origin. Complete the parallelogram by drawing parallel copies of each vector from the tip of the other. The enclosed region has area 13 square units. This area equals the absolute value of the determinant of the matrix formed by the two vectors as columns.
Sections 6.4 – 6.6
The \(3 \times 3\) Determinant and Cofactor Expansion
Row Expansion · Volume Interpretation · Minors · Cofactors · General Method
Section 6.4
Determinants of \(3 \times 3\) Matrices
For a \(3 \times 3\) matrix, the determinant is computed by expansion along the first row. For
\[ A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}, \]
Expansion Along the First Row
\[\det(A) = a_{11}\begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix} - a_{12}\begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix} + a_{13}\begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix}\]
Each \(2 \times 2\) determinant is the determinant of the submatrix formed by deleting the row and column containing that entry. The coefficients alternate in sign: \(+,\ -,\ +\).
Rule of Sarrus — A Memory Device for \(3 \times 3\) Only
For \(3 \times 3\) matrices, some students use the Rule of Sarrus: write the first two columns again to the right of the matrix, then sum three downward-diagonal products and subtract three upward-diagonal products. This memory device only works for \(3 \times 3\) matrices. Do not apply it to \(4 \times 4\) or larger matrices.
Example 6.4.1
Computing a \(3 \times 3\) determinant by row expansion
- ProblemCompute \(\det(A)\) for \(A = \begin{pmatrix} 2 & -1 & 3 \\ 0 & 4 & -2 \\ 1 & -3 & 5 \end{pmatrix}\).
- ThinkExpand along the first row. For each entry, compute the \(2 \times 2\) determinant of the submatrix formed by deleting that entry's row and column. Apply the sign pattern \(+, -, +\).
- Calculate
Delete row 1, col 1: det[4, -2; -3, 5] = (4)(5) - (-2)(-3) = 20 - 6 = 14
Delete row 1, col 2: det[0, -2; 1, 5] = (0)(5) - (-2)(1) = 0 + 2 = 2
Delete row 1, col 3: det[0, 4; 1, -3] = (0)(-3) - (4)(1) = 0 - 4 = -4
\[\det(A) = (+1)(2)(14) + (-1)(-1)(2) + (+1)(3)(-4) = 28 + 2 - 12 = 18\]
- Check\(28 + 2 = 30\), then \(30 - 12 = 18\). The computation checks out.
- Interpret\(\det(A) = 18\). The transformation scales volumes by 18 and preserves orientation. Matrix \(A\) is invertible.
Answer: det(A) = 18
Example 6.4.2
A \(3 \times 3\) determinant equal to zero
- ProblemCompute \(\det(A)\) for \(A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}\).
- Calculate
\[\det\begin{pmatrix} 5 & 6 \\ 8 & 9 \end{pmatrix} = 45 - 48 = -3\]
\[\det\begin{pmatrix} 4 & 6 \\ 7 & 9 \end{pmatrix} = 36 - 42 = -6\]
\[\det\begin{pmatrix} 4 & 5 \\ 7 & 8 \end{pmatrix} = 32 - 35 = -3\]
\[\det(A) = (1)(-3) - (2)(-6) + (3)(-3) = -3 + 12 - 9 = 0\]
- Interpret\(\det(A) = 0\). The matrix is singular. Notice that row 3 minus row 2 equals row 2 minus row 1 — the rows form an arithmetic progression and are linearly dependent. Linearly dependent rows always produce a zero determinant.
Answer: det(A) = 0 (singular matrix)
Section 6.5
Volume Interpretation
The \(3 \times 3\) determinant has the same geometric meaning in three dimensions that the \(2 \times 2\) determinant has in two: it is a volume scaling factor. Consider the three column vectors of a \(3 \times 3\) matrix \(A\). These three vectors define a parallelepiped — a three-dimensional solid analogous to a parallelogram. The volume of that parallelepiped equals \(|\det(A)|\).
The unit cube has volume 1. When you apply \(A\) to the unit cube, the image is a parallelepiped with volume \(|\det(A)|\). If \(|\det(A)| = 0\), the three column vectors are coplanar or collinear, and the parallelepiped collapses to a flat region with zero volume.
Example 6.5.1
Volume scaling by a diagonal matrix
- ProblemFind the volume of the parallelepiped produced when \(A = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{pmatrix}\) is applied to the unit cube.
- ThinkThe determinant of a diagonal matrix is the product of its diagonal entries.
- Calculate\[\det(A) = (2)(3)(4) = 24\]
- CheckThis matrix stretches \(x\) by 2, \(y\) by 3, and \(z\) by 4, giving a \(2 \times 3 \times 4 = 24\) rectangular box. Matches the determinant.
- InterpretThe transformation scales volume by a factor of 24. The matrix is invertible.
Answer: Volume = 24 cubic units
Section 6.6
Cofactor Expansion as an Introduction
The row expansion from Section 6.4 is a special case of a general method called cofactor expansion (or Laplace expansion). This method defines the determinant for any square matrix and allows computation by expanding along any row or column.
Minors and Cofactors
The minor \(M_{ij}\) of a matrix \(A\) is the determinant of the submatrix formed by deleting row \(i\) and column \(j\). The cofactor \(C_{ij}\) is the minor with a sign attached:
\[C_{ij} = (-1)^{i+j} M_{ij}\]
The factor \((-1)^{i+j}\) produces a checkerboard sign pattern. For a \(3 \times 3\) matrix:
Position \((i,j)\) carries a positive sign when \(i+j\) is even, and a negative sign when \(i+j\) is odd. Keep this grid in mind when performing cofactor expansion.
Cofactor Expansion — General Formula
Expanding along row \(i\):
\[\det(A) = a_{i1}C_{i1} + a_{i2}C_{i2} + \cdots + a_{in}C_{in}\]
Expanding along column \(j\):
\[\det(A) = a_{1j}C_{1j} + a_{2j}C_{2j} + \cdots + a_{nj}C_{nj}\]
Strategy: choose the row or column with the most zeros. Zero entries contribute nothing to the sum and dramatically simplify computation.
Example 6.6.1
Cofactor expansion along a convenient row
- ProblemCompute \(\det(A)\) for \(A = \begin{pmatrix} 5 & 0 & 0 \\ 2 & -3 & 1 \\ -1 & 4 & 2 \end{pmatrix}\).
- ThinkExpanding along row 1 is easiest because two of its three entries are zero.
- Calculate
\[\det(A) = 5 \cdot C_{11} + 0 + 0 = 5 \cdot (+1) \cdot \det\begin{pmatrix} -3 & 1 \\ 4 & 2 \end{pmatrix}\]
\[= 5 \cdot [(-3)(2) - (1)(4)] = 5 \cdot [-6 - 4] = 5 \cdot (-10) = -50\]
- Interpret\(\det(A) = -50\). The transformation scales volume by 50 and reverses orientation. The matrix is invertible.
Answer: det(A) = −50
Example 6.6.2
Cofactor expansion along a column
- ProblemCompute \(\det(A)\) for \(A = \begin{pmatrix} 3 & 0 & -1 \\ 2 & 5 & 0 \\ 0 & 1 & 4 \end{pmatrix}\).
- ThinkColumn 2 has a zero at the top, so expanding along column 2 simplifies computation.
- Calculate
\[\det(A) = 0 \cdot C_{12} + 5 \cdot C_{22} + 1 \cdot C_{32}\]
\[C_{22} = (+1)\det\begin{pmatrix} 3 & -1 \\ 0 & 4 \end{pmatrix} = (12 - 0) = 12\]
\[C_{32} = (-1)\det\begin{pmatrix} 3 & -1 \\ 2 & 0 \end{pmatrix} = (-1)(0+2) = -2\]
\[\det(A) = 0 + 5(12) + 1(-2) = 60 - 2 = 58\]
- Interpret\(\det(A) = 58\). The transformation is invertible, scales volumes by 58, and preserves orientation.
Answer: det(A) = 58
Sections 6.7 – 6.10
Properties, Row Operations, and Structural Connections
Eleven Key Properties · Row Reduction Strategy · Invertibility · Linear Dependence
Section 6.7
Determinant Properties
Determinants obey a rich set of algebraic properties. Learning them allows you to simplify determinant calculations, recognize zero determinants without full expansion, and reason about matrices structurally.
Property 1 — Identity Matrix
\(\det(I_n) = 1\). The identity transformation preserves all areas and volumes, scales by 1, and preserves orientation.
Property 2 — Transpose
\(\det(A^T) = \det(A)\). The determinant is unchanged by transposition. Any property that holds for rows also holds for columns.
Property 3 — Row Interchange (Sign Reversal)
Swapping two rows multiplies the determinant by \(-1\). Every row swap flips orientation, reversing the sign.
Property 4 — Row Scaling
Multiplying every entry in one row by scalar \(k\) multiplies the determinant by \(k\). A common factor in a single row can be factored out of the determinant.
Property 5 — Row Addition (No Change)
Adding a scalar multiple of one row to another row leaves the determinant unchanged. This is why row reduction by addition preserves \(\det(A)\).
Property 6 — Scalar Matrix Multiplication
For an \(n \times n\) matrix: \(\det(kA) = k^n \det(A)\). Scaling the entire matrix by \(k\) scales all \(n\) rows, multiplying the determinant by \(k^n\) — not by \(k\).
Property 7 — Multiplicative Property
\(\det(AB) = \det(A) \cdot \det(B)\). The scaling factors of successive transformations multiply together. This is one of the most powerful determinant identities.
Property 8 — Inverse
If \(A\) is invertible: \(\det(A^{-1}) = \dfrac{1}{\det(A)}\). This follows from \(\det(A)\cdot\det(A^{-1}) = \det(I) = 1\).
Property 9 — Zero Row or Column
If any row or column is entirely zero, then \(\det(A) = 0\). The transformation collapses at least one direction entirely.
Property 10 — Equal Rows or Columns
If any two rows (or two columns) are identical, then \(\det(A) = 0\). Swapping identical rows changes the sign but not the matrix, so \(\det(A) = -\det(A)\), forcing \(\det(A) = 0\).
Property 11 — Triangular Matrices
If \(A\) is upper or lower triangular, \(\det(A) = a_{11} \cdot a_{22} \cdot a_{33} \cdots a_{nn}\) — the product of diagonal entries. This is an extremely useful computational shortcut.
Example 6.7.1
Using properties to evaluate a determinant
- GivenSuppose \(\det(A) = 61\) for a certain \(3 \times 3\) matrix. Find the determinant of \(B\) where \(B\) is formed by: (a) doubling row 2; (b) swapping rows 1 and 3; (c) adding 3 times row 1 to row 3.
- Calculate
(a) Doubling row 2 → det(B) = 2 · 61 = 122
(b) Swapping rows 1 and 3 → det(B) = −61
(c) Adding a row multiple → det(B) = 61 (unchanged)
- InterpretThese properties reflect geometric facts: scaling a row scales area by the same factor; swapping rows reverses orientation; adding a row multiple does not change the overall scaling.
Answers: (a) 122, (b) −61, (c) 61
Example 6.7.2
Determinant of a product and scalar multiples
- Given\(\det(A) = 3\), \(\det(B) = -2\), \(A\) is \(3 \times 3\). Find \(\det(AB)\), \(\det(A^2)\), and \(\det(3A)\).
- Calculate
\[\det(AB) = \det(A) \cdot \det(B) = (3)(-2) = -6\]
\[\det(A^2) = [\det(A)]^2 = 9\]
\[\det(3A) = 3^3 \cdot \det(A) = 27 \cdot 3 = 81\]
- Interpret\(\det(AB) = -6\): composing the transformations scales volume by 6 and flips orientation. \(\det(A^2) = 9\): applying \(A\) twice scales volume by \(3^2\). \(\det(3A) = 81\): scaling the whole \(3 \times 3\) matrix by 3 multiplies the determinant by \(3^3\).
Answers: det(AB) = −6, det(A²) = 9, det(3A) = 81
Section 6.8
Row Operations and Determinants
| Row Operation | Effect on Determinant |
|---|
| Swap two rows | Multiply \(\det\) by \(-1\) |
| Multiply a row by \(k \neq 0\) | Multiply \(\det\) by \(k\) |
| Add \(c\) times one row to another | No change to \(\det\) |
Using these rules, you can row-reduce a matrix to upper triangular form (tracking all swaps and scalings), then use Property 11: the determinant of the triangular result is the product of its diagonal entries. This approach is especially efficient for larger matrices where direct cofactor expansion becomes tedious.
Strategy — Determinant by Row Reduction
1. Perform row operations to reduce \(A\) to upper triangular form \(U\).
2. Track each swap (multiplies det by \(-1\)) and each row scaling (multiplies det by the scale factor).
3. Compute \(\det(U)\) = product of diagonal entries.
4. Adjust for the row operations tracked in step 2.
Example 6.8.1
Determinant by row reduction
- ProblemCompute \(\det(A)\) for \(A = \begin{pmatrix} 0 & 2 & 1 \\ 3 & -1 & 4 \\ 1 & 0 & -2 \end{pmatrix}\).
- Step 1The top-left entry is zero. Swap row 1 and row 3 (multiply det by \(-1\)):
\[A \to \begin{pmatrix} 1 & 0 & -2 \\ 3 & -1 & 4 \\ 0 & 2 & 1 \end{pmatrix} \quad \text{factor: } -1\]
- Step 2Eliminate below col 1 pivot. \(R_2 \to R_2 - 3R_1\) (no change to det):
\[\begin{pmatrix} 1 & 0 & -2 \\ 0 & -1 & 10 \\ 0 & 2 & 1 \end{pmatrix} \quad \text{factor: } -1\]
- Step 3Eliminate below col 2 pivot. \(R_3 \to R_3 + 2R_2\) (no change to det):
\[\begin{pmatrix} 1 & 0 & -2 \\ 0 & -1 & 10 \\ 0 & 0 & 21 \end{pmatrix} \quad \text{factor: } -1\]
- Step 4Upper triangular. Diagonal product: \((1)(-1)(21) = -21\).
- Step 5Apply tracking factor: \(\det(A) = (-1)(-21) = 21\).
- InterpretThe transformation scales volumes by 21 and preserves orientation. Matrix \(A\) is invertible.
Answer: det(A) = 21
Section 6.9
Determinants and Invertibility
Core Theorem — Determinant and Invertibility
\[A \text{ is invertible} \iff \det(A) \neq 0\]
This is one of the most important results in linear algebra. It connects the algebraic property of invertibility directly to a geometric measurement of the transformation.
This theorem is part of a larger equivalence result known as the Invertible Matrix Theorem. Every condition in the table below is equivalent to every other condition for a square \(n \times n\) matrix \(A\):
| Equivalent Conditions for an \(n \times n\) Matrix \(A\) |
|---|
| \(A\) is invertible |
| \(\det(A) \neq 0\) |
| The system \(A\mathbf{x} = \mathbf{b}\) has a unique solution for every \(\mathbf{b}\) |
| The only solution to \(A\mathbf{x} = \mathbf{0}\) is \(\mathbf{x} = \mathbf{0}\) |
| The columns of \(A\) are linearly independent |
| The rows of \(A\) reduce to \(n\) pivot positions |
| \(A\) is row-equivalent to \(I_n\) |
| The transformation \(T(\mathbf{x}) = A\mathbf{x}\) is one-to-one and onto |
Example 6.9.1
Testing invertibility via the determinant
- ProblemDetermine whether each matrix is invertible.
\[P = \begin{pmatrix} 4 & 7 \\ 2 & 6 \end{pmatrix}, \quad Q = \begin{pmatrix} 1 & -2 & 3 \\ 0 & 0 & 5 \\ 2 & -4 & 6 \end{pmatrix}\]
- Calculate
\[\det(P) = (4)(6) - (7)(2) = 24 - 14 = 10 \neq 0 \quad \Rightarrow \text{invertible}\]
For \(Q\): row 3 is exactly twice row 1 — proportional rows give \(\det(Q) = 0\) — not invertible.
- InterpretMatrix \(P\) is invertible: its transformation of the plane is reversible and scales areas by 10. Matrix \(Q\) is singular: its transformation collapses three-dimensional space, and the system \(Q\mathbf{x} = \mathbf{b}\) either has no solution or infinitely many for most \(\mathbf{b}\).
Answers: P is invertible; Q is not invertible
Section 6.10
Determinants and Linear Dependence
Closely related to invertibility is linear dependence. The determinant detects whether the rows (or columns) of a matrix are linearly dependent:
Determinant and Linear Dependence
\[\det(A) = 0 \iff \text{the rows of } A \text{ are linearly dependent} \iff \text{the columns of } A \text{ are linearly dependent}\]
Conversely, \(\det(A) \neq 0\) implies both the rows and the columns are linearly independent. This provides a fast independence check for small matrices: compute the determinant without row reduction.
Example 6.10.1
Detecting dependence with determinants
- ProblemAre the vectors \(\mathbf{v}_1 = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}\), \(\mathbf{v}_2 = \begin{pmatrix} 2 \\ 4 \\ 6 \end{pmatrix}\), \(\mathbf{v}_3 = \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}\) linearly independent?
- RepresentForm the matrix with these vectors as columns:
\[A = \begin{pmatrix} 1 & 2 & -1 \\ 2 & 4 & 0 \\ 3 & 6 & 1 \end{pmatrix}\]
- CalculateColumn 2 is exactly twice column 1 (\(2\mathbf{v}_1 = \mathbf{v}_2\)). Two proportional columns give \(\det(A) = 0\).
- InterpretThe vectors are linearly dependent. Specifically, \(\mathbf{v}_2 = 2\mathbf{v}_1\), so \(\mathbf{v}_2\) adds no new direction to the set.
Answer: Linearly dependent — det(A) = 0
Sections 6.11 – 6.14
Cramer's Rule, Applications, Mistakes, and Calculus Readiness
Cramer's Rule · Geometry and Modeling · Common Errors · Bridge to Chapter 7
Section 6.11
Cramer's Rule as an Introduction
Cramer's Rule is a formula for solving a linear system \(A\mathbf{x} = \mathbf{b}\) using determinants, when \(A\) is invertible. For a system of \(n\) equations in \(n\) unknowns, define \(A_i\) as the matrix obtained from \(A\) by replacing column \(i\) with the vector \(\mathbf{b}\). Then:
Cramer's Rule
\[x_i = \frac{\det(A_i)}{\det(A)}, \quad i = 1, 2, \ldots, n\]
This requires \(\det(A) \neq 0\) — the system must have a unique solution. Cramer's Rule is theoretically elegant and beautiful. It expresses each unknown directly as a ratio of determinants. However, for systems larger than \(3 \times 3\), it is far less computationally efficient than row reduction.
When to Use Cramer's Rule
Cramer's Rule is most useful for \(2 \times 2\) and \(3 \times 3\) systems, for symbolic calculations where entries are letters rather than numbers, and for theoretical proofs. For practical computation on larger systems, use row reduction instead.
Example 6.11.1
Cramer's Rule for a \(2 \times 2\) system
- ProblemSolve: \(\begin{cases} 3x + 2y = 7 \\ x - y = 1 \end{cases}\)
- Represent
\[A = \begin{pmatrix} 3 & 2 \\ 1 & -1 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 7 \\ 1 \end{pmatrix}\]
- Calculate
\[\det(A) = (3)(-1) - (2)(1) = -5\]
\[A_1 = \begin{pmatrix} 7 & 2 \\ 1 & -1 \end{pmatrix}, \quad \det(A_1) = -7 - 2 = -9\]
\[A_2 = \begin{pmatrix} 3 & 7 \\ 1 & 1 \end{pmatrix}, \quad \det(A_2) = 3 - 7 = -4\]
\[x = \frac{-9}{-5} = \frac{9}{5}, \quad y = \frac{-4}{-5} = \frac{4}{5}\]
- Check\(3\!\left(\tfrac{9}{5}\right) + 2\!\left(\tfrac{4}{5}\right) = \tfrac{27+8}{5} = \tfrac{35}{5} = 7\) ✓ \(\tfrac{9}{5} - \tfrac{4}{5} = \tfrac{5}{5} = 1\) ✓
- InterpretThe unique solution is \(x = \tfrac{9}{5}\), \(y = \tfrac{4}{5}\). The nonzero determinant \(-5\) confirms exactly one solution exists.
Answer: x = 9/5, y = 4/5
Section 6.12
Determinants in Geometry and Modeling
Area of a Triangle
Given three points \(P_1 = (x_1, y_1)\), \(P_2 = (x_2, y_2)\), \(P_3 = (x_3, y_3)\), the area of the triangle with those vertices is:
\[\text{Area} = \frac{1}{2} \left| \det\begin{pmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{pmatrix} \right|\]
Other Applications
| Field | How Determinants Are Used |
|---|
| Computer Graphics | Check sign of det to detect orientation reversal in 2D transformations; validate matrices before applying to objects |
| Physics / Cross Product | The cross product \(\mathbf{u} \times \mathbf{v}\) is computed via a \(3 \times 3\) determinant with \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) in row 1; its magnitude equals the area of the parallelogram spanned by \(\mathbf{u}\) and \(\mathbf{v}\) |
| Robotics | Rotation matrices always have \(\det = 1\); a det of \(-1\) signals a reflection; engineers check determinants to validate rigid-body transformations |
| Economics | The Leontief input-output model requires \(\det(I - A) \neq 0\) to guarantee a unique equilibrium production vector |
Example 6.12.1
Area of a triangle using a determinant
- ProblemFind the area of the triangle with vertices \(P_1 = (1, 2)\), \(P_2 = (4, 6)\), \(P_3 = (3, 1)\).
- Represent
\[M = \begin{pmatrix} 1 & 2 & 1 \\ 4 & 6 & 1 \\ 3 & 1 & 1 \end{pmatrix}\]
- CalculateExpand along column 3 (all entries are 1):
\[C_{13} = (+1)\det\begin{pmatrix} 4 & 6 \\ 3 & 1 \end{pmatrix} = 4 - 18 = -14\]
\[C_{23} = (-1)\det\begin{pmatrix} 1 & 2 \\ 3 & 1 \end{pmatrix} = (-1)(1 - 6) = 5\]
\[C_{33} = (+1)\det\begin{pmatrix} 1 & 2 \\ 4 & 6 \end{pmatrix} = 6 - 8 = -2\]
\[\det(M) = -14 + 5 + (-2) = -11\]
\[\text{Area} = \tfrac{1}{2}|-11| = 5.5 \text{ square units}\]
- InterpretThe triangle has area 5.5 square units. The negative determinant indicates the vertices are listed in clockwise order.
Answer: Area = 5.5 square units
Section 6.13
Common Determinant Mistakes
Mistake 1 — Confusing the determinant with the matrix
det(A) = (the matrix itself)
The determinant is a single scalar number. Writing det(A) = \(\begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}\) is meaningless. det(A) = (3)(4) − (1)(2) = 10.
Mistake 2 — Wrong sign pattern in cofactor expansion
Using + signs for all positions in cofactor expansion
The sign is \((-1)^{i+j}\). Always write the sign grid before expanding: +, −, + on row 1; −, +, − on row 2; +, −, + on row 3.
Mistake 3 — Assuming det(A + B) = det(A) + det(B)
det(A + B) = det(A) + det(B)
This is false in general. The multiplicative property det(AB) = det(A)·det(B) holds, but additive splitting does not.
Mistake 4 — Scaling the whole matrix vs. one row
det(2A) = 2·det(A) for a 3×3 matrix
For an n×n matrix: det(kA) = kn·det(A). For a 3×3 matrix, det(2A) = 2³·det(A) = 8·det(A).
Mistake 5 — Scaling the determinant when adding rows
Adding c times row i to row j multiplies the determinant by c
Row addition (Rj → Rj + c·Ri) does not change the determinant. Only row swaps and row scalings change it.
Mistake 6 — Forgetting to track row swaps during reduction
Performing two row swaps without adjusting the sign
Each swap multiplies the determinant by −1. Two swaps restore the sign; three swaps flip it again. Always track every swap.
Mistake 7 — Applying the Rule of Sarrus to larger matrices
Using the diagonal-product shortcut on a 4×4 matrix
The Rule of Sarrus only works for 3×3 matrices. For 4×4 or larger, use cofactor expansion or row reduction.
Mistake 8 — Confusing |det(A)| with det(A) for areas
Area = det(A) even when det(A) is negative
Area and volume are always nonneg. The scaling factor is |det(A)|. The sign of det(A) carries orientation information — it does not make area negative.
Mistake 9 — Computing determinants of non-square matrices
Applying the 2×2 formula to a 2×3 matrix
Determinants are only defined for square (n×n) matrices. A 2×3 or 4×2 matrix has no determinant.
Mistake 10 — Using wrong formula: ad + bc instead of ad − bc
det[[a,b],[c,d]] = ad + bc
The 2×2 determinant is always ad − bc. The off-diagonal product bc is subtracted, not added.
Section 6.14
Preparing for Linear Transformations
You have now built a solid foundation in determinants: what they measure, how to compute them, how row operations affect them, and how they connect to invertibility and linear dependence. Chapter 7 takes the next major step: studying linear transformations as functions from vectors to vectors.
In Chapter 7, you will see that every matrix defines a linear transformation and every linear transformation from \(\mathbb{R}^n\) to \(\mathbb{R}^m\) is represented by a matrix. The determinant will reappear in that context as a measurement of how the transformation changes area or volume, and you will see the concrete geometric realizations of rotations, reflections, projections, and shears — all of which you can now characterize by their determinants.
Looking further ahead, determinants return in Chapter 12 when you compute eigenvalues using the characteristic polynomial \(\det(A - \lambda I) = 0\). Every eigenvalue computation begins with a determinant.
Bridge to Calculus
The Jacobian Determinant
In multivariable calculus, the Jacobian determinant appears in change-of-variables formulas for multiple integrals. It is exactly the two- or three-dimensional generalization of the scaling concept developed in this chapter. When you substitute coordinates in a double integral — converting from rectangular to polar coordinates, for example — the Jacobian is the determinant of the matrix of partial derivatives of the transformation. The geometric foundation you have built here directly supports that concept in your future calculus courses.
MGU Library Connection
- Appendix H: Determinant Formula Reference — \(2 \times 2\) formula, \(3 \times 3\) cofactor expansion, sign pattern grid, property table, Cramer's Rule, triangular matrix shortcut
- Appendix R: Calculus Readiness Bridge Guide — preview of the Jacobian determinant in multivariable calculus
- Chapter 7: Linear Transformations — determinants reappear as area/volume scaling factors of rotations, reflections, shears, and projections
- Chapter 12: Eigenvalues and Eigenvectors — the characteristic polynomial \(\det(A - \lambda I) = 0\) requires determinant computation for every eigenvalue problem
- Physics chapters: the cross product \(\mathbf{u} \times \mathbf{v}\) is computed via determinant notation
- Computer Graphics lessons: orientation detection via the sign of the determinant in rendering pipelines
Chapter 6 — Back Matter
Chapter Summary
- Sec 6.1The determinant is a scalar that measures area scaling (2D) or volume scaling (3D). Positive: preserves orientation. Negative: reverses orientation. Zero: collapses dimension; matrix is singular.
- Sec 6.2For a \(2 \times 2\) matrix, \(\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc\). Written with vertical bars. Notation must not be confused with absolute value.
- Sec 6.3The area of the parallelogram spanned by the column vectors of \(A\) is \(|\det(A)|\). The unit square transforms into this parallelogram. The sign of the determinant encodes orientation.
- Sec 6.4The \(3 \times 3\) determinant is computed by expanding along the first row with alternating signs \(+, -, +\). The Rule of Sarrus is a \(3 \times 3\)-only memory device.
- Sec 6.5The volume of the parallelepiped defined by the column vectors of a \(3 \times 3\) matrix is \(|\det(A)|\). The unit cube maps to this parallelepiped. Sign encodes 3D orientation.
- Sec 6.6Cofactor expansion generalizes to any row or column: \(\det(A) = \sum_j a_{ij} C_{ij}\) where \(C_{ij} = (-1)^{i+j} M_{ij}\). Always expand along the row or column with the most zeros.
- Sec 6.7Eleven key properties: identity, transpose, row swap (sign flip), row scaling, row addition (no change), scalar mult (\(k^n\)), product rule, inverse reciprocal, zero row, equal rows, triangular matrix product.
- Sec 6.8Row reduction computes determinants efficiently: track swaps (\(\times -1\)) and scalings (\(\times k\)); row additions are free. Determinant of triangular result is the diagonal product.
- Sec 6.9\(A\) is invertible iff \(\det(A) \neq 0\). This is part of the Invertible Matrix Theorem, which lists eight equivalent conditions for invertibility of a square matrix.
- Sec 6.10\(\det(A) = 0\) iff the rows (and columns) of \(A\) are linearly dependent. Proportional or linearly combined rows/columns always produce a zero determinant.
- Sec 6.11Cramer's Rule: \(x_i = \det(A_i)/\det(A)\) where \(A_i\) is \(A\) with column \(i\) replaced by \(\mathbf{b}\). Elegant for \(2 \times 2\) and \(3 \times 3\) systems; not efficient for larger ones.
- Sec 6.12Determinants compute triangle areas, cross products, rotation checks in robotics, orientation in graphics, and economic equilibrium conditions. The Jacobian determinant generalizes these ideas in calculus.
- Sec 6.13Ten common mistakes: det is a scalar not a matrix; wrong sign pattern; additive det fallacy; \(k^n\) vs. \(k\); row addition changes nothing; track all swaps; Sarrus only for \(3 \times 3\); use \(|\det|\) for area; square matrices only; subtract not add off-diagonal.
- Sec 6.14Determinants prepare students for Chapter 7 (linear transformations), Chapter 12 (eigenvalues via characteristic polynomial), and multivariable calculus (Jacobian determinant for change of variables).
Key Terms Review
determinant
A scalar associated with every square matrix that measures area or volume scaling and encodes orientation and invertibility information.
minor \(M_{ij}\)
The determinant of the submatrix obtained by deleting row \(i\) and column \(j\) from \(A\).
cofactor \(C_{ij}\)
The signed minor: \(C_{ij} = (-1)^{i+j} M_{ij}\). The sign follows the checkerboard pattern of the entry's position.
cofactor expansion
A method for computing \(\det(A)\) by expanding along any row or column, summing the products of entries with their cofactors.
orientation
The sense of "clockwise" vs. "counterclockwise" (2D) or "right-handed" vs. "left-handed" (3D), preserved when \(\det(A) > 0\), reversed when \(\det(A) < 0\).
singular matrix
A square matrix with \(\det(A) = 0\); not invertible; its transformation collapses dimension.
Cramer's Rule
For an invertible system \(A\mathbf{x} = \mathbf{b}\): \(x_i = \det(A_i)/\det(A)\), where \(A_i\) is \(A\) with column \(i\) replaced by \(\mathbf{b}\).
triangular matrix
A matrix with all zeros above (lower) or below (upper) the main diagonal. Its determinant equals the product of diagonal entries.
area scaling factor
The factor by which a \(2 \times 2\) transformation changes all areas. Equal to \(|\det(A)|\).
volume scaling factor
The factor by which a \(3 \times 3\) transformation changes all volumes. Equal to \(|\det(A)|\).
multiplicative property
The identity \(\det(AB) = \det(A)\cdot\det(B)\). The scaling factors of successive transformations multiply.
Invertible Matrix Theorem
A collection of equivalent conditions for invertibility, including \(\det(A) \neq 0\), column independence, unique solutions, and row equivalence to \(I_n\).
parallelepiped
A three-dimensional solid with six parallelogram faces; the 3D analog of a parallelogram. Its volume equals \(|\det(A)|\) when its edges are the column vectors of \(A\).
multilinearity
The property that the determinant is linear in each row (or column) separately when all other rows are held fixed.
Concept Review Questions
- Explain in your own words what the determinant of a \(2 \times 2\) matrix measures geometrically. Give a concrete example.
- What does a determinant of zero tell you about a matrix transformation? What does it mean for the system \(A\mathbf{x} = \mathbf{b}\)?
- How does the sign of the determinant relate to orientation? Give one example where the determinant is negative and describe the transformation.
- What is a minor? What is a cofactor? How do they differ?
- When performing cofactor expansion, which row or column should you prefer? Why?
- List three properties of determinants. Give a brief geometric explanation for each one.
- State the connection between determinants and invertibility. What is the Invertible Matrix Theorem?
- Why is Cramer's Rule used more often for theoretical purposes than for computation?
- Explain why the Rule of Sarrus cannot be used for \(4 \times 4\) matrices.
- How does the \(3 \times 3\) determinant relate to the cross product in physics?
Determinant Computation Practice
Compute the determinant of each matrix. State whether the matrix is invertible.
- \(\begin{pmatrix} 3 & 1 \\ 2 & 5 \end{pmatrix}\)
- \(\begin{pmatrix} -4 & 7 \\ 2 & -3 \end{pmatrix}\)
- \(\begin{pmatrix} 0 & 8 \\ -2 & 6 \end{pmatrix}\)
- \(\begin{pmatrix} \frac{1}{2} & 3 \\ -1 & 4 \end{pmatrix}\)
- \(\begin{pmatrix} 5 & -5 \\ -3 & 3 \end{pmatrix}\)
- \(\begin{pmatrix} a & b \\ 2a & 2b \end{pmatrix}\) (general \(a, b\) — what do you notice?)
Compute the determinant of each matrix using cofactor expansion. Choose your row or column strategically.
- \(\begin{pmatrix} 1 & 2 & 0 \\ -1 & 3 & 1 \\ 2 & 0 & 4 \end{pmatrix}\)
- \(\begin{pmatrix} 2 & -1 & 0 \\ 0 & 3 & 5 \\ 1 & 4 & -2 \end{pmatrix}\)
- \(\begin{pmatrix} 4 & 0 & 0 \\ 2 & -3 & 0 \\ 1 & 5 & 7 \end{pmatrix}\)
- \(\begin{pmatrix} 0 & 1 & -1 \\ 3 & 0 & 2 \\ -1 & 4 & 0 \end{pmatrix}\)
- \(\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\)
- \(\begin{pmatrix} 2 & 4 & -6 \\ 1 & 2 & -3 \\ 5 & 3 & 1 \end{pmatrix}\) (examine rows 1 and 2 first)
- Given \(\det(A) = 5\) for a \(3 \times 3\) matrix, find: (a) \(\det(A^T)\), (b) \(\det(2A)\), (c) \(\det(A^{-1})\), (d) \(\det(-A)\).
- If \(\det(A) = 4\) and \(\det(B) = -3\), find \(\det(AB)\), \(\det(BA)\), and \(\det(A^2B)\).
- Suppose you swap rows 1 and 2 of a matrix with \(\det = 7\), then multiply row 3 by \(-2\), then add row 1 to row 2. What is the new determinant?
- Vectors \(\mathbf{v}_1 = \begin{pmatrix} 5 \\ 2 \end{pmatrix}\) and \(\mathbf{v}_2 = \begin{pmatrix} -1 \\ 3 \end{pmatrix}\) define a parallelogram based at the origin. Find its area.
- The unit square is mapped by \(A = \begin{pmatrix} 3 & 1 \\ 0 & 2 \end{pmatrix}\). What is the area of the image? Is orientation preserved?
- Find the area of the triangle with vertices \((0, 0)\), \((4, 1)\), \((2, 5)\) using the determinant formula.
- Find the area of the triangle with vertices \((1, 3)\), \((4, 1)\), \((2, 7)\).
- Three vectors \(\mathbf{v}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}\), \(\mathbf{v}_2 = \begin{pmatrix} 0 \\ 2 \\ 0 \end{pmatrix}\), \(\mathbf{v}_3 = \begin{pmatrix} 0 \\ 0 \\ 3 \end{pmatrix}\) define a parallelepiped. What is its volume?
- Use the determinant to determine whether each matrix is invertible. Do not compute the inverse.
(a) \(\begin{pmatrix} 6 & -4 \\ -3 & 2 \end{pmatrix}\)
(b) \(\begin{pmatrix} 7 & 2 \\ 1 & -1 \end{pmatrix}\)
(c) \(\begin{pmatrix} 2 & 1 & -1 \\ 0 & 3 & 2 \\ 4 & 2 & -2 \end{pmatrix}\)
(d) \(\begin{pmatrix} 1 & 0 & 0 \\ 2 & 5 & 0 \\ 3 & -1 & 4 \end{pmatrix}\)
- Find all values of \(k\) for which \(\begin{pmatrix} k & 3 \\ 2 & k \end{pmatrix}\) is not invertible.
- Find all values of \(t\) for which \(\begin{pmatrix} t & 1 & 0 \\ 0 & t & 1 \\ 0 & 0 & t \end{pmatrix}\) is not invertible.
Application Problems
- Computer Graphics. A game engine uses \(A = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}\) to rotate sprites. (a) Compute \(\det(A)\) using \(\cos^2\theta + \sin^2\theta = 1\). (b) What does this tell you about area scaling and orientation of the rotation?
- Robotics. A transformation matrix has \(\det(A) = -1\). An engineer claims this represents a pure rotation. Is the engineer correct? Explain.
- Economics. In a Leontief input-output model, the matrix \((I - A)\) must be invertible to guarantee a unique equilibrium production vector. If \(\det(I - A) = 0\), what does that imply about the model?
- Physics Readiness. Use the determinant formula to compute the cross product \(\mathbf{u} \times \mathbf{v}\) where \(\mathbf{u} = \begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix}\) and \(\mathbf{v} = \begin{pmatrix} 0 \\ 3 \\ 1 \end{pmatrix}\). What is the geometric meaning of the magnitude of the result?
- Cramer's Rule. Solve the system \(\begin{cases} 4x - y = 5 \\ 2x + 3y = 1 \end{cases}\) using Cramer's Rule.
- Land Area. A land parcel is modeled as a parallelogram. The two defining edge vectors are \(\mathbf{v}_1 = \begin{pmatrix} 120 \\ 30 \end{pmatrix}\) and \(\mathbf{v}_2 = \begin{pmatrix} 0 \\ 80 \end{pmatrix}\) (in meters). Find the area of the parcel.
Error Analysis
Each problem shows a student's work containing an error. Identify the error and show the correct solution.
Problem 1
det[[3,2],[1,4]] = 3·2 − 1·4 = 6 − 4 = 2
What did the student do wrong? What is the correct answer?
Problem 2
If det(A) = 6 for a 2×2 matrix, then det(3A) = 3·6 = 18
What is the correct answer and why?
Problem 3
Expanding along row 1: det(A) = 2·det[4,5;2,0] + 0·(...) + (−1)·det[3,4;−1,2]
For \(A = \begin{pmatrix} 2 & 0 & -1 \\ 3 & 4 & 5 \\ -1 & 2 & 0 \end{pmatrix}\), identify the sign error. Write the correct expansion.
Problem 4
"Since det(A) = det(B) = 5, we have det(A + B) = 10."
Is this correct? Provide a counterexample or a clear explanation.
Problem 5
A student uses the Rule of Sarrus to compute the determinant of a 4×4 matrix.
Explain why this is incorrect and describe the proper method.
Chapter 6 Checkpoint
Chapter 6 Checkpoint
Complete without assistance — assesses all key skills and concepts from Chapter 6
Section 1 — Computation
- C6.1Compute \(\det\begin{pmatrix} -2 & 5 \\ 4 & -3 \end{pmatrix}\).
- C6.2Compute \(\det\begin{pmatrix} 3 & -1 & 2 \\ 0 & 4 & 1 \\ 5 & 0 & -2 \end{pmatrix}\) by expanding along the most convenient row or column. State which you chose.
- C6.3A matrix \(A\) has \(\det(A) = -8\). Find: (a) \(\det(A^T)\), (b) \(\det(2A)\) if \(A\) is \(3 \times 3\), (c) \(\det(A^{-1})\).
Section 2 — Invertibility and Dependence
- C6.4Determine whether \(\begin{pmatrix} 6 & 9 \\ 4 & 6 \end{pmatrix}\) is invertible. Explain using the determinant.
- C6.5Without full row reduction, determine whether the columns of \(\begin{pmatrix} 2 & -4 & 8 \\ 1 & -2 & 4 \\ 3 & 0 & -1 \end{pmatrix}\) are linearly independent. Justify your answer.
Section 3 — Geometric Interpretation
- C6.6Vectors \(\mathbf{v}_1 = \begin{pmatrix} 4 \\ 1 \end{pmatrix}\) and \(\mathbf{v}_2 = \begin{pmatrix} -2 \\ 3 \end{pmatrix}\) are adjacent sides of a parallelogram based at the origin. Find the area.
- C6.7Compute the area of the triangle with vertices \((2, 1)\), \((5, 4)\), and \((1, 6)\).
Section 4 — Cramer's Rule and Properties
- C6.8Use Cramer's Rule to solve: \(\begin{cases} 2x + 5y = 4 \\ x + 3y = 1 \end{cases}\)
- C6.9A matrix undergoes the following row operations: rows 1 and 2 are swapped; then row 3 is multiplied by \(-3\); then 5 times row 1 is added to row 2. If the original determinant was 4, what is the final determinant?
Section 5 — Interpretation
- C6.10A transformation matrix \(A\) has \(\det(A) = -3\). Describe what this tells you about: (a) the area/volume scaling factor, (b) whether the transformation preserves or reverses orientation, (c) whether the transformation is invertible.
Section 6 — Error Analysis
- C6.11A student simplifies \(\det(5A)\) for a \(3 \times 3\) matrix with \(\det(A) = 2\) and writes \(\det(5A) = 5 \cdot 2 = 10\). What is the error? What is the correct answer?
- C6.12A student performs cofactor expansion on a \(3 \times 3\) matrix and assigns all \(2 \times 2\) minors a positive sign regardless of position. Explain the error and show how to apply the correct sign pattern.
Answers
- Answers to all practice problems and checkpoint items may be placed in the Answer Key (Appendix back matter of the full textbook edition).