The Central Idea
For a square matrix \(A\), the determinant is a scalar written
\[ \boxed{\det(A)} \]or sometimes
\[ \boxed{|A|}. \]The determinant measures how the corresponding linear transformation scales oriented \(n\)-dimensional volume.
Most importantly,
\[ \boxed{\det(A)\neq0\iff A\text{ is invertible}}. \]Determinants Apply to Square Matrices
The ordinary determinant is defined for square matrices:
\[ n\times n. \]Examples include
\[ 1\times1,\qquad2\times2,\qquad3\times3, \]and higher square dimensions.
A rectangular \(m\times n\) matrix with \(m\neq n\) does not have an ordinary determinant.
The \(1\times1\) Determinant
For the matrix
\[ A= \begin{bmatrix} a \end{bmatrix}, \]the determinant is simply
\[ \boxed{\det(A)=a}. \]This is the base case underlying cofactor expansion.
The \(2\times2\) Determinant
For
\[ A= \begin{bmatrix} a&b\\ c&d \end{bmatrix}, \]the determinant is
\[ \boxed{\det(A)=ad-bc}. \]A useful memory pattern is
\[ \boxed{\text{main diagonal product}-\text{other diagonal product}}. \]Example: Compute a \(2\times2\) Determinant
Let
\[ A= \begin{bmatrix} 3&2\\ 5&4 \end{bmatrix}. \]Then
\[ \det(A) = 3(4)-2(5) = 12-10. \]Therefore,
\[ \boxed{\det(A)=2}. \]Example: A Zero Determinant
Consider
\[ A= \begin{bmatrix} 1&2\\ 2&4 \end{bmatrix}. \]Then
\[ \det(A)=1(4)-2(2)=0. \]The second column is twice the first, so the columns are linearly dependent.
The zero determinant records this loss of independent direction.
Determinant and Area in \(\mathbb{R}^2\)
Let the columns of
\[ A= \begin{bmatrix} a&b\\ c&d \end{bmatrix} \]be two vectors in the plane.
The parallelogram they determine has area
\[ \boxed{|\det(A)|}. \]The sign carries orientation information, while the absolute value gives ordinary geometric area.
Orientation in Two Dimensions
A positive determinant preserves orientation.
A negative determinant reverses orientation.
A zero determinant collapses area to zero.
Thus the sign and magnitude of the determinant encode different kinds of geometric information.
Determinant as an Area-Scaling Factor
Suppose a linear transformation
\[ T(x)=Ax \]acts on a region in \(\mathbb{R}^2\).
If the original region has area \(S\), the transformed region has area
\[ \boxed{|\det(A)|S}. \]The determinant therefore measures global area scaling for linear maps.
Example: Scaling Transformation
Let
\[ A= \begin{bmatrix} 3&0\\ 0&2 \end{bmatrix}. \]Horizontal lengths are scaled by \(3\), and vertical lengths by \(2\).
Therefore areas are scaled by
\[ 3\cdot2=6. \]Indeed,
\[ \boxed{\det(A)=6}. \]Example: Reflection
Reflection across the \(x\)-axis has matrix
\[ A= \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix}. \]Its determinant is
\[ \boxed{-1}. \]Area magnitude is preserved, but orientation is reversed.
Example: Rotation
A rotation matrix is
\[ R_\theta= \begin{bmatrix} \cos\theta&-\sin\theta\\ \sin\theta&\cos\theta \end{bmatrix}. \]Its determinant is
\[ \cos^2\theta+\sin^2\theta=1. \]Therefore,
\[ \boxed{\det(R_\theta)=1}. \]Rotation preserves both area and orientation.
Example: Shear
The shear matrix
\[ A= \begin{bmatrix} 1&k\\ 0&1 \end{bmatrix} \]has determinant
\[ \boxed{1}. \]A shear changes shape but preserves area.
The \(3\times3\) Determinant
For
\[ A= \begin{bmatrix} a&b&c\\ d&e&f\\ g&h&i \end{bmatrix}, \]one expansion along the first row is
\[ \boxed{ \det(A) = a(ei-fh) - b(di-fg) + c(dh-eg)}. \]Minors
The minor \(M_{ij}\) is the determinant of the matrix obtained by deleting row \(i\) and column \(j\).
For example, if
\[ A= \begin{bmatrix} a&b&c\\ d&e&f\\ g&h&i \end{bmatrix}, \]then
\[ M_{12} = \det \begin{bmatrix} d&f\\ g&i \end{bmatrix} = di-fg. \]Cofactors
The cofactor \(C_{ij}\) is
\[ \boxed{C_{ij}=(-1)^{i+j}M_{ij}}. \]The signs follow the checkerboard pattern
\[ \begin{bmatrix} +&-&+\\ -&+&-\\ +&-&+ \end{bmatrix}. \]Cofactor Expansion
The determinant may be expanded along any row or any column.
Expanding along row \(i\),
\[ \boxed{ \det(A) = a_{i1}C_{i1} +a_{i2}C_{i2} +\cdots+ a_{in}C_{in}}. \]Expanding along a row or column containing many zeros usually saves work.
Example: Cofactor Expansion
Let
\[ A= \begin{bmatrix} 1&2&0\\ 3&4&5\\ 0&6&7 \end{bmatrix}. \]Expanding along the first row,
\[ \det(A) = 1 \det \begin{bmatrix} 4&5\\ 6&7 \end{bmatrix} - 2 \det \begin{bmatrix} 3&5\\ 0&7 \end{bmatrix}. \]Therefore,
\[ \det(A) = (28-30)-2(21) = -2-42. \]Hence
\[ \boxed{\det(A)=-44}. \]Choose the Easiest Row or Column
Cofactor expansion gives the same determinant regardless of which row or column is used.
For practical computation, choose a row or column containing many zeros or simple entries.
Strategic expansion reduces arithmetic and lowers the chance of sign errors.
Determinant and Volume in \(\mathbb{R}^3\)
If the columns of a \(3\times3\) matrix are three vectors in \(\mathbb{R}^3\), then
\[ \boxed{|\det(A)|} \]is the volume of the parallelepiped formed by those vectors.
If the determinant is zero, the vectors lie in a lower-dimensional configuration and enclose no three-dimensional volume.
Determinant in Higher Dimensions
In \(\mathbb{R}^n\), the absolute value
\[ |\det(A)| \]measures the factor by which the linear transformation scales \(n\)-dimensional volume.
The geometric object generalizing length, area, and volume is sometimes called hypervolume.
Determinant Zero Means Dimension Collapse
If
\[ \det(A)=0, \]then \(A\) collapses some nonzero direction.
The image has dimension less than \(n\), the columns are linearly dependent, and the transformation cannot be inverted.
Determinant Nonzero Means Full Dimension
If
\[ \det(A)\neq0, \]then the columns of \(A\) remain linearly independent.
The transformation maps \(n\)-dimensional volume to nonzero \(n\)-dimensional volume.
Consequently, the matrix has full rank.
The Invertible Matrix Connection
For an \(n\times n\) matrix \(A\), the following are equivalent:
- \(\det(A)\neq0\);
- \(A\) is invertible;
- \(\operatorname{rank}(A)=n\);
- the columns of \(A\) are linearly independent;
- the columns of \(A\) span \(\mathbb{R}^n\);
- \(Ax=0\) has only the trivial solution;
- \(Ax=b\) has a unique solution for every \(b\in\mathbb{R}^n\).
Determinant of the Identity Matrix
The identity transformation preserves every vector and every volume.
Therefore,
\[ \boxed{\det(I_n)=1}. \]Determinant of a Diagonal Matrix
For
\[ D= \begin{bmatrix} d_1&0&\cdots&0\\ 0&d_2&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&d_n \end{bmatrix}, \]the determinant is
\[ \boxed{\det(D)=d_1d_2\cdots d_n}. \]Determinant of a Triangular Matrix
If \(A\) is upper triangular or lower triangular, then
\[ \boxed{\det(A)=a_{11}a_{22}\cdots a_{nn}}. \]The determinant is simply the product of the diagonal entries.
This makes elimination a powerful determinant-computation method.
Effect of Swapping Two Rows
Interchanging two rows reverses orientation.
Therefore,
\[ \boxed{\det(B)=-\det(A)} \]when \(B\) is obtained from \(A\) by one row swap.
Effect of Scaling a Row
If one row is multiplied by a scalar \(c\), then the determinant is multiplied by \(c\):
\[ \boxed{\det(B)=c\det(A)}. \]Scaling two rows by \(c\) and \(d\) would multiply the determinant by \(cd\).
Effect of Row Replacement
Adding a scalar multiple of one row to another row does not change the determinant.
\[ \boxed{ R_i\leftarrow R_i+cR_j \quad\Longrightarrow\quad \det(B)=\det(A)}. \]This property makes elimination especially useful.
Computing a Determinant by Elimination
To compute a large determinant efficiently:
- use row replacement to create zeros below pivots;
- track every row swap;
- track every row scaling;
- reduce to triangular form;
- multiply the diagonal entries;
- correct for swaps and scalings.
Example: Elimination Method
Let
\[ A= \begin{bmatrix} 1&2&3\\ 2&5&8\\ 1&0&1 \end{bmatrix}. \]Apply
\[ R_2\leftarrow R_2-2R_1, \qquad R_3\leftarrow R_3-R_1. \]Row replacement does not change the determinant:
\[ \begin{bmatrix} 1&2&3\\ 0&1&2\\ 0&-2&-2 \end{bmatrix}. \]Then apply
\[ R_3\leftarrow R_3+2R_2, \]giving
\[ \begin{bmatrix} 1&2&3\\ 0&1&2\\ 0&0&2 \end{bmatrix}. \]Therefore,
\[ \boxed{\det(A)=1\cdot1\cdot2=2}. \]Equal Rows Force a Zero Determinant
If two rows of a square matrix are identical, then
\[ \boxed{\det(A)=0}. \]Swapping those rows leaves the matrix unchanged, but a row swap must change the determinant's sign.
The only scalar equal to its own negative is \(0\).
Proportional Rows Force a Zero Determinant
If one row is a scalar multiple of another, the rows are linearly dependent.
Therefore,
\[ \boxed{\det(A)=0}. \]The same conclusion holds for linearly dependent columns.
A Zero Row Forces a Zero Determinant
If a square matrix contains a zero row, then
\[ \boxed{\det(A)=0}. \]A zero row prevents full rank and indicates that the transformation cannot preserve full dimension.
Determinant of a Product
For square matrices \(A\) and \(B\) of the same size,
\[ \boxed{\det(AB)=\det(A)\det(B)}. \]Geometrically, the volume-scaling factors multiply when transformations are composed.
Why the Product Rule Makes Sense
Suppose \(A\) scales volume by a factor of
\[ |\det(A)| \]and \(B\) scales volume by
\[ |\det(B)|. \]Applying \(B\) and then \(A\) scales volume by the product of those factors.
Orientation signs multiply as well.
Determinant of an Inverse
If \(A\) is invertible, then
\[ AA^{-1}=I. \]Taking determinants,
\[ \det(A)\det(A^{-1})=1. \]Therefore,
\[ \boxed{ \det(A^{-1}) = \frac{1}{\det(A)}}. \]Determinant of a Power
For an integer \(k\geq1\),
\[ A^k= \underbrace{AA\cdots A}_{k\text{ factors}}. \]Therefore,
\[ \boxed{\det(A^k)=\det(A)^k}. \]Determinant of a Scalar Multiple
For an \(n\times n\) matrix,
\[ \boxed{\det(cA)=c^n\det(A)}. \]Multiplying the entire matrix by \(c\) scales each of its \(n\) rows by \(c\), producing \(n\) determinant factors.
Determinant of a Transpose
Transposing a square matrix does not change its determinant:
\[ \boxed{\det(A^T)=\det(A)}. \]Determinant properties therefore apply symmetrically to rows and columns.
Columns and Rows Play Symmetric Roles
The determinant can be expanded along rows or columns.
Dependence among rows forces determinant zero, and dependence among columns also forces determinant zero.
This symmetry is reflected by
\[ \det(A^T)=\det(A). \]Determinant and Basis Formation
Let
\[ A= \begin{bmatrix} \vert&&\vert\\ v_1&\cdots&v_n\\ \vert&&\vert \end{bmatrix} \]contain \(n\) vectors in \(\mathbb{R}^n\) as columns.
Then
\[ \boxed{ \{v_1,\ldots,v_n\}\text{ is a basis for }\mathbb{R}^n \iff \det(A)\neq0}. \]Example: Basis Test in \(\mathbb{R}^2\)
Consider
\[ v_1= \begin{bmatrix} 1\\ 2 \end{bmatrix}, \qquad v_2= \begin{bmatrix} 3\\ 1 \end{bmatrix}. \]Form
\[ A= \begin{bmatrix} 1&3\\ 2&1 \end{bmatrix}. \]Then
\[ \det(A)=1-6=-5\neq0. \]Therefore the vectors form a basis for \(\mathbb{R}^2\).
Determinant and Coordinate Changes
A change-of-basis matrix must be invertible.
Therefore its determinant must be nonzero.
The determinant also records the oriented volume scaling between the two coordinate frames.
The Adjugate Formula
For an invertible square matrix,
\[ \boxed{ A^{-1} = \frac{1}{\det(A)} \operatorname{adj}(A)}. \]The adjugate is obtained by forming the cofactor matrix and transposing it.
This formula makes the role of the nonzero determinant in inversion explicit.
The \(2\times2\) Inverse Formula
For
\[ A= \begin{bmatrix} a&b\\ c&d \end{bmatrix}, \]if
\[ ad-bc\neq0, \]then
\[ \boxed{ A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d&-b\\ -c&a \end{bmatrix}}. \]Determinants and Cramer's Rule
Determinants can be used to solve an invertible square system
\[ Ax=b. \]Cramer's Rule expresses each unknown as a quotient of determinants.
For example,
\[ \boxed{ x_i= \frac{\det(A_i)}{\det(A)}}, \]where \(A_i\) is obtained from \(A\) by replacing column \(i\) with \(b\).
When Cramer's Rule Is Useful
Cramer's Rule is conceptually important and useful for small symbolic systems.
For large numerical systems, elimination or factorization methods are generally more efficient.
The rule is especially valuable because it reveals another direct connection between determinants and unique solvability.
Determinants and Linear Dependence
For a square matrix,
\[ \boxed{ \det(A)=0 \iff \text{the columns are linearly dependent}}. \]The same is true for the rows.
The determinant can therefore serve as a compact dependence test when the number of vectors equals the ambient dimension.
Determinant and the Null Space
If
\[ \det(A)=0, \]then
\[ Ax=0 \]has a nontrivial solution.
Thus
\[ \boxed{\operatorname{Null}(A)\neq\{0\}}. \]Determinant and Rank
For an \(n\times n\) matrix,
\[ \boxed{ \det(A)\neq0 \iff \operatorname{rank}(A)=n}. \]If the determinant is zero, the rank is less than \(n\).
Determinants and Eigenvalues
The next major use of determinants comes from asking when
\[ Av=\lambda v. \]Rearranging gives
\[ (A-\lambda I)v=0. \]A nonzero solution \(v\) exists only when
\[ \boxed{\det(A-\lambda I)=0}. \]This equation leads directly to eigenvalues.
Common Error: Using the \(2\times2\) Rule on Larger Matrices
The shortcut
\[ ad-bc \]applies only to a \(2\times2\) matrix.
Larger determinants require cofactor expansion, elimination, or another valid determinant method.
Common Error: Forgetting Cofactor Signs
Cofactor expansion uses the alternating sign pattern
\[ \begin{bmatrix} +&-&+\\ -&+&-\\ +&-&+ \end{bmatrix}. \]Omitting the negative signs produces incorrect determinants.
Common Error: Ignoring Row Swaps During Elimination
Every row swap multiplies the determinant by \(-1\).
If two swaps occur, the two sign changes cancel.
Always track the parity of the number of row interchanges.
Common Error: Ignoring Row Scaling
If a row is divided by \(5\) during elimination, the determinant of the new matrix is one-fifth of the old determinant.
Row scaling must be recorded and corrected when the final determinant is recovered.
Common Error: Treating Row Replacement Like Row Scaling
The operation
\[ R_i\leftarrow R_i+cR_j \]does not change the determinant.
This differs fundamentally from multiplying an entire row by \(c\).
Common Error: Using \(\det(A+B)=\det(A)+\det(B)\)
In general,
\[ \boxed{ \det(A+B)\neq\det(A)+\det(B)}. \]The determinant is not a linear function of the entire matrix considered as one object.
Common Error: Thinking Negative Determinant Means Negative Area
Ordinary area and volume are nonnegative.
The magnitude
\[ |\det(A)| \]gives the geometric scaling factor.
A negative determinant indicates orientation reversal, not negative physical volume.
Common Error: Assuming Determinant Measures Matrix Size
The determinant is not a general measure of how large the entries of a matrix are.
A matrix can contain large entries and still have determinant zero because its columns are dependent.
Determinants measure oriented volume scaling and invertibility structure.
A Reliable Procedure for a \(2\times2\) Determinant
- Identify \(a,b,c,d\) in the matrix.
- Compute the main diagonal product \(ad\).
- Compute the other diagonal product \(bc\).
- Subtract: \(ad-bc\).
- Interpret zero versus nonzero when relevant.
A Reliable Procedure for Cofactor Expansion
- Choose a row or column, preferably one containing many zeros.
- For each selected entry, delete its row and column.
- Compute the resulting minor determinant.
- Apply the sign factor \((-1)^{i+j}\).
- Multiply the entry by its cofactor.
- Add all resulting terms.
A Reliable Procedure Using Elimination
- Use row replacement freely; it leaves the determinant unchanged.
- Record every row interchange; each reverses the sign.
- Avoid row scaling when possible, or record its factor carefully.
- Reduce the matrix to triangular form.
- Multiply the diagonal entries.
- Correct for swaps and row scalings.
A Reliable Procedure for an Invertibility Test
- Confirm that the matrix is square.
- Compute its determinant.
- If \(\det(A)\neq0\), conclude that \(A\) is invertible.
- If \(\det(A)=0\), conclude that \(A\) is singular.
- Connect the result to rank, independence, and system solvability as needed.
Example: Full \(3\times3\) Workflow
Compute
\[ \det \begin{bmatrix} 2&1&0\\ 3&-1&4\\ 0&5&2 \end{bmatrix}. \]Expand along the first row:
\[ 2 \det \begin{bmatrix} -1&4\\ 5&2 \end{bmatrix} - 1 \det \begin{bmatrix} 3&4\\ 0&2 \end{bmatrix}. \]Compute the minors:
\[ (-1)(2)-4(5)=-22, \] \[ 3(2)-4(0)=6. \]Therefore,
\[ \det(A)=2(-22)-6=-50. \]Hence
\[ \boxed{\det(A)=-50}. \]Example: Full Geometric Interpretation
Suppose
\[ A= \begin{bmatrix} 2&1\\ 1&3 \end{bmatrix}. \]Then
\[ \det(A)=2(3)-1(1)=5. \]The corresponding linear transformation multiplies every planar area by
\[ \boxed{5}. \]Because the determinant is positive, orientation is preserved.
Why Determinants Matter in Multidimensional Systems
Determinants summarize whether a square linear system retains all of its independent dimensions.
A nonzero determinant signals full rank, invertibility, unique solvability, and nonzero volume scaling. A zero determinant signals dependence, singularity, and dimensional collapse.
Determinants also appear in coordinate transformations, Jacobians, eigenvalue equations, stability analysis, geometry, differential equations, and multivariable integration.
Practice 1: \(2\times2\) Determinant
Compute
\[ \det \begin{bmatrix} 4&3\\ 2&5 \end{bmatrix}. \]Practice 2: Invertibility
Determine whether
\[ A= \begin{bmatrix} 2&6\\ 1&3 \end{bmatrix} \]is invertible.
Practice 3: Area Scaling
A linear transformation has matrix
\[ A= \begin{bmatrix} 3&0\\ 0&4 \end{bmatrix}. \]By what factor are planar areas scaled?
Practice 4: Orientation
A transformation has determinant \(-7\). What does the sign tell you about orientation?
Practice 5: Triangular Matrix
Compute
\[ \det \begin{bmatrix} 2&4&1\\ 0&-3&5\\ 0&0&6 \end{bmatrix}. \]Practice 6: Row Swap
If \(\det(A)=9\) and \(B\) is obtained from \(A\) by swapping two rows once, what is \(\det(B)\)?
Practice 7: Row Scaling
If \(\det(A)=5\) and one row is multiplied by \(4\), what is the determinant of the resulting matrix?
Practice 8: Row Replacement
If \(\det(A)=-3\) and
\[ R_2\leftarrow R_2+7R_1, \]what happens to the determinant?
Practice 9: Product Rule
If
\[ \det(A)=2 \]and
\[ \det(B)=-5, \]find \(\det(AB)\).
Practice 10: Inverse Determinant
If
\[ \det(A)=8, \]find
\[ \det(A^{-1}). \]Practice 11: Basis Test
Do the vectors
\[ \begin{bmatrix} 1\\ 2 \end{bmatrix}, \qquad \begin{bmatrix} 2\\ 4 \end{bmatrix} \]form a basis for \(\mathbb{R}^2\)?
Practice 12: Cofactor Expansion
Compute
\[ \det \begin{bmatrix} 1&0&2\\ 3&4&5\\ 0&6&7 \end{bmatrix}. \]Practice Answers
Practice 1
\[ \det \begin{bmatrix} 4&3\\ 2&5 \end{bmatrix} = 4(5)-3(2) = 20-6 = \boxed{14}. \]Practice 2
\[ \det(A) = 2(3)-6(1) = 0. \]Therefore,
\[ \boxed{A\text{ is not invertible}}. \]Practice 3
\[ |\det(A)| = |3(4)| = \boxed{12}. \]Areas are multiplied by \(12\).
Practice 4
The negative sign means the transformation
\[ \boxed{\text{reverses orientation}}. \]The area or volume scaling factor is \(7\).
Practice 5
The matrix is triangular, so multiply the diagonal entries:
\[ 2(-3)(6)=\boxed{-36}. \]Practice 6
One row swap reverses the sign:
\[ \boxed{\det(B)=-9}. \]Practice 7
Scaling one row by \(4\) multiplies the determinant by \(4\):
\[ \boxed{20}. \]Practice 8
Row replacement does not change the determinant:
\[ \boxed{-3}. \]Practice 9
\[ \det(AB) = \det(A)\det(B) = 2(-5) = \boxed{-10}. \]Practice 10
\[ \det(A^{-1}) = \frac{1}{\det(A)} = \boxed{\frac18}. \]Practice 11
Form the matrix
\[ A= \begin{bmatrix} 1&2\\ 2&4 \end{bmatrix}. \]Then
\[ \det(A)=4-4=0. \]Therefore the vectors are dependent and
\[ \boxed{\text{do not form a basis for }\mathbb{R}^2}. \]Practice 12
Expand along the first row:
\[ \det(A) = 1 \det \begin{bmatrix} 4&5\\ 6&7 \end{bmatrix} + 2 \det \begin{bmatrix} 3&4\\ 0&6 \end{bmatrix}. \]Thus
\[ \det(A) = (28-30)+2(18) = -2+36 = \boxed{34}. \]What to Remember
For a square matrix \(A\), the determinant is a scalar that encodes invertibility and oriented volume scaling.
- For \(2\times2\), \(\det(A)=ad-bc\).
- Cofactor expansion works along any row or column.
- Triangular determinants are products of diagonal entries.
- A row swap reverses the determinant's sign.
- Scaling one row scales the determinant by the same factor.
- Row replacement leaves the determinant unchanged.
- \(\det(AB)=\det(A)\det(B)\).
- \(\det(A^{-1})=1/\det(A)\) when \(A\) is invertible.
- \(\det(A^T)=\det(A)\).
- \(\det(A)\neq0\) exactly when \(A\) is invertible.
- \(|\det(A)|\) is the \(n\)-dimensional volume-scaling factor.
The central structural test is:
\[ \boxed{ \det(A)=0 \iff \text{the transformation loses an independent dimension}}. \]Where We Go Next
Determinants reveal when a matrix becomes singular. They also let us detect special scalars for which
\[ A-\lambda I \]becomes singular.
The next lesson, Eigenvalues and Eigenvectors, uses
\[ \det(A-\lambda I)=0 \]to identify directions that a linear transformation preserves while merely scaling them.