The Central Idea
Suppose a vector \(\mathbf{v}\) is first transformed by matrix \(A\).
The result is
\[ A\mathbf{v}. \]Now suppose matrix \(B\) acts on that result.
We obtain
\[ B(A\mathbf{v}). \]Matrix multiplication lets us combine the two transformations:
\[ \boxed{ B(A\mathbf{v}) = (BA)\mathbf{v} }. \]Composition Becomes Multiplication
In function language, applying \(A\) and then \(B\) is a composition.
In matrix language, that composition is represented by
\[ \boxed{BA}. \]The rightmost matrix acts first.
This order convention is one of the most important habits in transformation algebra.
Why the Rightmost Matrix Acts First
Begin with
\[ \mathbf{v}. \]Apply \(A\):
\[ \mathbf{v} \longmapsto A\mathbf{v}. \]Then apply \(B\):
\[ A\mathbf{v} \longmapsto B(A\mathbf{v}). \]By associativity,
\[ B(A\mathbf{v}) = (BA)\mathbf{v}. \]Thus \(A\) appears nearest the vector because it acts first.
Matrix Multiplication Is Not Entry-by-Entry Multiplication
Suppose
\[ A= \begin{bmatrix} a&b\\ c&d \end{bmatrix} \]and
\[ B= \begin{bmatrix} e&f\\ g&h \end{bmatrix}. \]The product \(AB\) is not found by simply multiplying entries in matching positions.
Instead, each product entry comes from a row of \(A\) and a column of \(B\).
The Row-by-Column Rule
For
\[ A= \begin{bmatrix} a&b\\ c&d \end{bmatrix}, \qquad B= \begin{bmatrix} e&f\\ g&h \end{bmatrix}, \]the product is
\[ \boxed{ AB = \begin{bmatrix} ae+bg & af+bh\\ ce+dg & cf+dh \end{bmatrix} }. \]Where the Top-Left Entry Comes From
The top-left entry of \(AB\) uses the first row of \(A\):
\[ \begin{bmatrix} a&b \end{bmatrix} \]and the first column of \(B\):
\[ \begin{bmatrix} e\\ g \end{bmatrix}. \]Their row-column product is
\[ \boxed{ae+bg}. \]Where the Top-Right Entry Comes From
Use the first row of \(A\) and the second column of \(B\):
\[ \begin{bmatrix} a&b \end{bmatrix} \begin{bmatrix} f\\ h \end{bmatrix}. \]Multiply corresponding entries and add:
\[ \boxed{af+bh}. \]Where the Bottom-Left Entry Comes From
Use the second row of \(A\) and the first column of \(B\):
\[ \begin{bmatrix} c&d \end{bmatrix} \begin{bmatrix} e\\ g \end{bmatrix}. \]Therefore,
\[ \boxed{ce+dg}. \]Where the Bottom-Right Entry Comes From
Use the second row of \(A\) and the second column of \(B\):
\[ \begin{bmatrix} c&d \end{bmatrix} \begin{bmatrix} f\\ h \end{bmatrix}. \]Therefore,
\[ \boxed{cf+dh}. \]A Compact Entry Formula
For matrices of compatible dimensions, the entry in row \(i\), column \(j\) of the product \(AB\) is
\[ \boxed{ (AB)_{ij} = \sum_k a_{ik}b_{kj} }. \]The index \(k\) runs across a row of \(A\) and down a column of \(B\).
Example: Multiply Two \(2\times2\) Matrices
Let
\[ A= \begin{bmatrix} 1&2\\ 3&4 \end{bmatrix} \]and
\[ B= \begin{bmatrix} 5&6\\ 7&8 \end{bmatrix}. \]Then
\[ AB = \begin{bmatrix} 1(5)+2(7) & 1(6)+2(8)\\ 3(5)+4(7) & 3(6)+4(8) \end{bmatrix}. \]Complete the Matrix Product
Simplifying each entry gives
\[ AB = \begin{bmatrix} 19&22\\ 43&50 \end{bmatrix}. \]Therefore,
\[ \boxed{ AB = \begin{bmatrix} 19&22\\ 43&50 \end{bmatrix} }. \]Matrix Dimensions Must Be Compatible
Suppose \(A\) has dimensions
\[ m\times n \]and \(B\) has dimensions
\[ n\times p. \]Then \(AB\) is defined and has dimensions
\[ \boxed{ m\times p }. \]The inner dimensions must match.
The Inner and Outer Dimensions
The pattern
\[ (m\times n)(n\times p) \]can be remembered as:
- the inner dimensions \(n\) and \(n\) must agree;
- the outer dimensions \(m\) and \(p\) determine the product size.
Thus,
\[ \boxed{ (m\times n)(n\times p) = m\times p }. \]Example: Compatible Dimensions
A
\[ 2\times3 \]matrix can multiply a
\[ 3\times4 \]matrix.
The result has dimensions
\[ \boxed{2\times4}. \]Example: Incompatible Dimensions
A
\[ 2\times3 \]matrix cannot multiply a
\[ 2\times4 \]matrix in that order because the inner dimensions are
\[ 3 \]and
\[ 2. \]They do not match.
Matrix-Vector Multiplication Is the Same Rule
A column vector can be viewed as a matrix with one column.
For example,
\[ A= \begin{bmatrix} a&b\\ c&d \end{bmatrix}, \qquad \mathbf{v} = \begin{bmatrix} x\\ y \end{bmatrix}. \]Since the dimensions are
\[ (2\times2)(2\times1), \]the result is a \(2\times1\) vector.
Matrix-Vector Product Revisited
Applying the row-by-column rule gives
\[ \boxed{ A\mathbf{v} = \begin{bmatrix} ax+by\\ cx+dy \end{bmatrix} }. \]Thus matrix-vector multiplication is not a separate procedure. It is a special case of matrix multiplication.
Composition of Two Transformations
Let \(A\) and \(B\) represent two plane transformations.
If \(A\) acts first and \(B\) acts second, then
\[ \mathbf{v} \longmapsto A\mathbf{v} \longmapsto B(A\mathbf{v}). \]The combined transformation matrix is
\[ \boxed{BA}. \]Example: Scale, Then Rotate
Suppose we first scale uniformly by \(2\):
\[ S= \begin{bmatrix} 2&0\\ 0&2 \end{bmatrix}. \]Then rotate \(90^\circ\) counterclockwise:
\[ R= \begin{bmatrix} 0&-1\\ 1&0 \end{bmatrix}. \]The combined matrix is
\[ RS. \]Compute the Scale-Then-Rotate Matrix
\[ RS = \begin{bmatrix} 0&-1\\ 1&0 \end{bmatrix} \begin{bmatrix} 2&0\\ 0&2 \end{bmatrix}. \]Multiplying gives
\[ \boxed{ RS = \begin{bmatrix} 0&-2\\ 2&0 \end{bmatrix} }. \]Verify the Combined Transformation on a Vector
Let
\[ \mathbf{v} = \begin{bmatrix} 3\\ 1 \end{bmatrix}. \]Scale first:
\[ S\mathbf{v} = \begin{bmatrix} 6\\ 2 \end{bmatrix}. \]Then rotate:
\[ R(S\mathbf{v}) = \begin{bmatrix} -2\\ 6 \end{bmatrix}. \]Verify with the Product Matrix
Now use
\[ RS = \begin{bmatrix} 0&-2\\ 2&0 \end{bmatrix}. \]Then
\[ (RS)\mathbf{v} = \begin{bmatrix} 0&-2\\ 2&0 \end{bmatrix} \begin{bmatrix} 3\\ 1 \end{bmatrix} = \boxed{ \begin{bmatrix} -2\\ 6 \end{bmatrix} }. \]The sequential and combined calculations agree.
Some Transformations Commute
Uniform scaling commutes with rotation about the origin.
If
\[ S=kI, \]then
\[ RS = SR = kR. \]This is a special case, not the general rule.
Matrix Multiplication Is Generally Not Commutative
In general,
\[ \boxed{ AB\neq BA }. \]This reflects the fact that performing transformation \(B\) and then transformation \(A\) can differ from performing \(A\) and then \(B\).
Example: Reflection and Shear
Let
\[ F= \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix} \]represent reflection across the \(x\)-axis, and let
\[ H= \begin{bmatrix} 1&1\\ 0&1 \end{bmatrix} \]represent a horizontal shear.
Compute \(FH\)
\[ FH = \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix} \begin{bmatrix} 1&1\\ 0&1 \end{bmatrix}. \]Therefore,
\[ \boxed{ FH = \begin{bmatrix} 1&1\\ 0&-1 \end{bmatrix} }. \]Compute \(HF\)
\[ HF = \begin{bmatrix} 1&1\\ 0&1 \end{bmatrix} \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix}. \]Therefore,
\[ \boxed{ HF = \begin{bmatrix} 1&-1\\ 0&-1 \end{bmatrix} }. \]Since \(FH\neq HF\), the two transformation orders differ.
Reading Composition Order from a Product
In
\[ BA, \]matrix \(A\) acts first.
Matrix \(B\) acts second.
A useful reading habit is:
\[ \boxed{ BA = \text{“first }A,\text{ then }B\text{”} }. \]Three Transformations
Suppose \(A\) acts first, \(B\) second, and \(C\) third.
Then
\[ \mathbf{v} \longmapsto A\mathbf{v} \longmapsto B(A\mathbf{v}) \longmapsto C(B(A\mathbf{v})). \]The combined matrix is
\[ \boxed{CBA}. \]Matrix Multiplication Is Associative
Although multiplication is generally not commutative, it is associative:
\[ \boxed{ (AB)C = A(BC) }. \]This means we may group a chain of compatible matrix multiplications in different ways without changing the final product.
Why Associativity Matters for Transformations
Consider three transformations:
\[ C(B(A\mathbf{v})). \]Associativity lets us write
\[ (CBA)\mathbf{v} \]without ambiguity about grouping.
The sequence remains ordered, but the parentheses can be reorganized for efficient computation.
The Identity Matrix in Composition
For a square matrix \(A\),
\[ \boxed{ IA=AI=A }. \]The identity matrix therefore acts like “do nothing” in transformation composition.
Example: Identity Transformation
Let
\[ A= \begin{bmatrix} 2&1\\ 0&3 \end{bmatrix}. \]Then
\[ IA = \begin{bmatrix} 1&0\\ 0&1 \end{bmatrix} \begin{bmatrix} 2&1\\ 0&3 \end{bmatrix} = \boxed{ \begin{bmatrix} 2&1\\ 0&3 \end{bmatrix} }. \]Inverse Transformations
Some transformations can be undone by another transformation.
If a square matrix \(A\) has an inverse \(A^{-1}\), then
\[ \boxed{ A^{-1}A = AA^{-1} = I }. \]Applying a transformation and then its inverse returns the original vector.
Rotation Inverses
A rotation by angle \(\theta\) is undone by rotation through
\[ -\theta. \]Therefore,
\[ \boxed{ R_\theta^{-1} = R_{-\theta} }. \]Their composition is the identity:
\[ R_{-\theta}R_\theta=I. \]Reflections Are Their Own Inverses
Reflecting twice across the same line returns every point to its original position.
Therefore, for a reflection matrix \(F\),
\[ \boxed{ F^2=I }. \]Thus
\[ F^{-1}=F. \]Example: Reflection Squared
For reflection across the \(x\)-axis,
\[ F= \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix}. \]Then
\[ F^2 = \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix} \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix} = \boxed{ \begin{bmatrix} 1&0\\ 0&1 \end{bmatrix} }. \]Uniform Scaling Inverses
A nonzero uniform scaling by \(k\) is undone by scaling by
\[ \frac1k. \]If
\[ S_k=kI, \]then
\[ \boxed{ S_k^{-1} = S_{1/k} } \]when \(k\neq0\).
Some Transformations Cannot Be Undone
If a matrix collapses different vectors to the same output, information is lost.
For example,
\[ A= \begin{bmatrix} 1&0\\ 0&0 \end{bmatrix} \]sends
\[ (x,y)\mapsto(x,0). \]The original \(y\)-coordinate cannot be recovered from the output alone.
Composition and the Images of Basis Vectors
The columns of a matrix show the images of the standard basis vectors.
For a composition \(BA\), the columns show what happens after \(A\) sends the basis vectors somewhere and \(B\) transforms those images again.
This gives a geometric interpretation of matrix multiplication.
First Column of a Product
The first column of \(A\) is
\[ A\mathbf{i}. \]Applying \(B\) gives
\[ B(A\mathbf{i}) = (BA)\mathbf{i}. \]Therefore, the first column of \(BA\) is the transformed image of \(\mathbf{i}\).
Second Column of a Product
Likewise, the second column of \(A\) is
\[ A\mathbf{j}. \]Then
\[ B(A\mathbf{j}) = (BA)\mathbf{j}. \]So the second column of \(BA\) records the final image of \(\mathbf{j}\).
Composition of Rotations
Rotating by \(\alpha\) and then by \(\beta\) gives a total rotation of
\[ \alpha+\beta. \]In matrix form,
\[ \boxed{ R_\beta R_\alpha = R_{\alpha+\beta} }. \]Rotations about the same origin therefore compose by adding their angles.
Example: Two \(90^\circ\) Rotations
Let
\[ R= \begin{bmatrix} 0&-1\\ 1&0 \end{bmatrix}. \]Then
\[ R^2 = \begin{bmatrix} 0&-1\\ 1&0 \end{bmatrix} \begin{bmatrix} 0&-1\\ 1&0 \end{bmatrix}. \]Multiplying gives
\[ \boxed{ R^2 = \begin{bmatrix} -1&0\\ 0&-1 \end{bmatrix} }. \]Two \(90^\circ\) rotations equal one \(180^\circ\) rotation.
Repeated Matrix Multiplication
Applying the same transformation repeatedly produces matrix powers.
If \(A\) acts \(n\) times, the combined transformation is
\[ \boxed{ A^n }. \]Matrix powers therefore encode repeated spatial operations.
Example: Four Quarter-Turns
For the \(90^\circ\) rotation matrix \(R\),
\[ R^4 = I. \]Four quarter-turns total
\[ 360^\circ, \]returning every vector to its original position.
Composition of Scalings
If one uniform scaling multiplies lengths by \(a\) and a second multiplies them by \(b\), the combined scale factor is
\[ ab. \]In matrix form,
\[ \boxed{ S_bS_a = S_{ab} }. \]Composition of Axis-Aligned Scalings
Let
\[ A= \begin{bmatrix} a&0\\ 0&b \end{bmatrix} \]and
\[ B= \begin{bmatrix} c&0\\ 0&d \end{bmatrix}. \]Then
\[ \boxed{ BA = \begin{bmatrix} ac&0\\ 0&bd \end{bmatrix} }. \]Composition of Shears
Two horizontal shears with factors \(a\) and \(b\) compose to give another horizontal shear.
\[ \begin{bmatrix} 1&b\\ 0&1 \end{bmatrix} \begin{bmatrix} 1&a\\ 0&1 \end{bmatrix} = \boxed{ \begin{bmatrix} 1&a+b\\ 0&1 \end{bmatrix} }. \]Their shear factors add.
Composition Can Create a New Transformation Type
Combining standard transformations does not always produce a matrix that looks like one of the original families.
A reflection followed by a shear, for example, may produce a matrix whose geometry must be interpreted from its total action rather than from one familiar template.
Composition Preserves Linearity
If \(A\) and \(B\) are linear transformations, then their composition is also linear.
In matrix form,
\[ BA \]is itself a matrix representing another linear transformation.
Thus linear transformations are closed under composition.
Why the Product Matrix Is Useful
Once several transformations are combined into one matrix, we can:
- apply the entire sequence with one matrix-vector multiplication;
- analyze the total geometric effect;
- reuse the transformation efficiently on many vectors;
- compare transformation sequences;
- study inverses and repeated applications.
Applying a Combined Matrix to an Entire Figure
Suppose a polygon has vertices
\[ \mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_n. \]If the total transformation matrix is
\[ M=BA, \]then each transformed vertex is
\[ \boxed{ M\mathbf{v}_i }. \]We do not need to repeat the full transformation sequence separately for each vertex.
Matrix Multiplication and Coordinate Rules
Suppose
\[ A= \begin{bmatrix} a&b\\ c&d \end{bmatrix}. \]It represents the coordinate rule
\[ (x,y) \mapsto (ax+by,\;cx+dy). \]Multiplying transformation matrices composes such coordinate rules automatically.
Example: Compose Coordinate Rules
Let \(A\) send
\[ (x,y)\mapsto(2x,y), \]so
\[ A= \begin{bmatrix} 2&0\\ 0&1 \end{bmatrix}. \]Let \(B\) send
\[ (x,y)\mapsto(x+y,y), \]so
\[ B= \begin{bmatrix} 1&1\\ 0&1 \end{bmatrix}. \]Compute the Combined Coordinate Rule
Since \(A\) acts first and \(B\) second,
\[ BA = \begin{bmatrix} 1&1\\ 0&1 \end{bmatrix} \begin{bmatrix} 2&0\\ 0&1 \end{bmatrix}. \]Thus,
\[ BA = \begin{bmatrix} 2&1\\ 0&1 \end{bmatrix}. \]Therefore, the combined rule is
\[ \boxed{ (x,y)\mapsto(2x+y,y) }. \]Verifying the Coordinate Rule Directly
First apply \(A\):
\[ (x,y)\mapsto(2x,y). \]Then apply \(B\):
\[ (2x,y)\mapsto(2x+y,y). \]This matches the rule obtained from the product matrix.
Common Error: Multiplying Corresponding Entries
For ordinary matrix multiplication,
\[ \begin{bmatrix} a&b\\ c&d \end{bmatrix} \begin{bmatrix} e&f\\ g&h \end{bmatrix} \]is not
\[ \begin{bmatrix} ae&bf\\ cg&dh \end{bmatrix}. \]Matrix multiplication uses row-by-column sums.
Common Error: Reversing Transformation Order
If \(A\) acts first and \(B\) acts second, the combined matrix is
\[ \boxed{BA}, \]not \(AB\).
Read matrix products from right to left when interpreting transformation order.
Common Error: Assuming \(AB=BA\)
Matrix multiplication is not generally commutative.
Even when both products exist,
\[ AB \]and
\[ BA \]can represent different transformations.
Common Error: Ignoring Dimensions
Before multiplying matrices, check the inner dimensions.
A product
\[ AB \]exists only when the number of columns of \(A\) equals the number of rows of \(B\).
Common Error: Giving the Product the Wrong Size
If
\[ A \text{ is }m\times n \]and
\[ B \text{ is }n\times p, \]then
\[ AB \]is
\[ \boxed{m\times p}. \]The inner dimensions disappear; the outer dimensions remain.
Common Error: Confusing Associativity with Commutativity
Associativity says
\[ (AB)C=A(BC). \]It does not say
\[ AB=BA. \]Grouping may change, but order generally may not.
Common Error: Treating \(A^2\) as Entry-Wise Squaring
Matrix powers mean repeated matrix multiplication.
Thus,
\[ \boxed{ A^2=AA }, \]not a matrix formed by squaring every individual entry.
Common Error: Assuming Every Matrix Has an Inverse
Some transformations collapse dimensions or otherwise lose information.
Such matrices cannot be undone and therefore do not have inverses.
Invertibility must be established rather than assumed.
A Reliable Procedure for Multiplying Two Matrices
- Write the dimensions of both matrices.
- Verify that the inner dimensions match.
- Determine the dimensions of the product from the outer dimensions.
- Choose one row from the first matrix and one column from the second.
- Multiply corresponding entries.
- Add those products to form one output entry.
- Repeat for every row-column combination.
- Check signs and arithmetic.
A Reliable Procedure for Composing Transformations
- Write the transformations in the order they occur.
- Assign a matrix to each transformation.
- Place the first transformation nearest the vector on the right.
- Place each later transformation to its left.
- Multiply the matrices in that order.
- Apply the resulting product matrix to vectors or figures.
- Verify one sample input by carrying out the transformations separately.
A Reliable Procedure for Checking a Composition Product
- Select a simple test vector.
- Apply the original transformation sequence one step at a time.
- Apply the proposed product matrix directly to the same vector.
- Compare the results.
- If they differ, check matrix order and row-column arithmetic first.
Example: Full Matrix-Multiplication Workflow
Let
\[ A= \begin{bmatrix} 2&1\\ -1&3 \end{bmatrix}, \qquad B= \begin{bmatrix} 4&0\\ 2&1 \end{bmatrix}. \]Find \(BA\).
The dimensions are
\[ (2\times2)(2\times2), \]so the product is defined and will be \(2\times2\).
Complete the Full Matrix Product
\[ BA = \begin{bmatrix} 4&0\\ 2&1 \end{bmatrix} \begin{bmatrix} 2&1\\ -1&3 \end{bmatrix}. \]Compute each entry:
\[ BA = \begin{bmatrix} 4(2)+0(-1) & 4(1)+0(3)\\ 2(2)+1(-1) & 2(1)+1(3) \end{bmatrix}. \]Therefore,
\[ \boxed{ BA = \begin{bmatrix} 8&4\\ 3&5 \end{bmatrix} }. \]Example: Full Transformation-Composition Workflow
First reflect across the \(y\)-axis:
\[ F= \begin{bmatrix} -1&0\\ 0&1 \end{bmatrix}. \]Then rotate \(90^\circ\) counterclockwise:
\[ R= \begin{bmatrix} 0&-1\\ 1&0 \end{bmatrix}. \]The combined matrix is
\[ RF. \]Compute the Combined Transformation
\[ RF = \begin{bmatrix} 0&-1\\ 1&0 \end{bmatrix} \begin{bmatrix} -1&0\\ 0&1 \end{bmatrix}. \]Thus,
\[ \boxed{ RF = \begin{bmatrix} 0&-1\\ -1&0 \end{bmatrix} }. \]This is the standard reflection matrix across the line
\[ y=-x. \]Practice 1: Multiply Two Matrices
Find
\[ \begin{bmatrix} 1&3\\ 2&4 \end{bmatrix} \begin{bmatrix} 2&0\\ 5&1 \end{bmatrix}. \]Practice 2: Check Dimensions
Determine whether the product
\[ (3\times2)(2\times4) \]is defined, and if so, state the dimensions of the result.
Practice 3: Incompatible Dimensions
Determine whether
\[ (2\times3)(4\times2) \]is defined.
Practice 4: Transformation Order
If matrix \(A\) acts first and matrix \(B\) acts second, which product represents the combined transformation?
Practice 5: Compose Two Rotations
A figure is rotated \(30^\circ\) counterclockwise and then \(45^\circ\) counterclockwise about the origin.
What single rotation is equivalent to the composition?
Practice 6: Square a Reflection Matrix
Let
\[ F= \begin{bmatrix} -1&0\\ 0&1 \end{bmatrix}. \]Find
\[ F^2. \]Practice 7: Compose Two Uniform Scalings
A vector is first scaled by \(3\) and then by \(\frac12\).
Find the single equivalent scale factor.
Practice 8: Apply a Product Matrix
Let
\[ M= \begin{bmatrix} 2&-1\\ 1&3 \end{bmatrix}. \]Find
\[ M \begin{bmatrix} 4\\ 2 \end{bmatrix}. \]Practice 9: Compare \(AB\) and \(BA\)
Let
\[ A= \begin{bmatrix} 1&1\\ 0&1 \end{bmatrix}, \qquad B= \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix}. \]Compute \(AB\) and \(BA\) and determine whether they are equal.
Practice 10: Identify the Error
A student says that if \(A\) acts first and \(B\) acts second, the combined matrix must be \(AB\) because \(A\) was mentioned first.
Explain the error.
Practice Answers
Practice 1
\[ \begin{bmatrix} 1&3\\ 2&4 \end{bmatrix} \begin{bmatrix} 2&0\\ 5&1 \end{bmatrix} = \begin{bmatrix} 1(2)+3(5) & 1(0)+3(1)\\ 2(2)+4(5) & 2(0)+4(1) \end{bmatrix}. \]Therefore,
\[ \boxed{ \begin{bmatrix} 17&3\\ 24&4 \end{bmatrix} }. \]Practice 2
The inner dimensions match:
\[ (3\times\boxed{2})(\boxed{2}\times4). \]Therefore, the product is defined and has dimensions
\[ \boxed{3\times4}. \]Practice 3
The inner dimensions are \(3\) and \(4\), so they do not match.
Therefore, the product is not defined.
Practice 4
If \(A\) acts first and \(B\) second, the combined matrix is
\[ \boxed{BA}. \]Practice 5
Rotations about the same origin add:
\[ 30^\circ+45^\circ=75^\circ. \]Therefore, the equivalent transformation is
\[ \boxed{\text{a }75^\circ\text{ counterclockwise rotation}}. \]Practice 6
\[ F^2 = \begin{bmatrix} -1&0\\ 0&1 \end{bmatrix} \begin{bmatrix} -1&0\\ 0&1 \end{bmatrix} = \boxed{ \begin{bmatrix} 1&0\\ 0&1 \end{bmatrix} }. \]Practice 7
\[ 3\left(\frac12\right) = \boxed{\frac32}. \]The combined transformation scales every vector by \(\frac32\).
Practice 8
\[ \begin{bmatrix} 2&-1\\ 1&3 \end{bmatrix} \begin{bmatrix} 4\\ 2 \end{bmatrix} = \begin{bmatrix} 8-2\\ 4+6 \end{bmatrix} = \boxed{ \begin{bmatrix} 6\\ 10 \end{bmatrix} }. \]Practice 9
\[ AB = \begin{bmatrix} 1&1\\ 0&1 \end{bmatrix} \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix} = \begin{bmatrix} 1&-1\\ 0&-1 \end{bmatrix}. \]But
\[ BA = \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix} \begin{bmatrix} 1&1\\ 0&1 \end{bmatrix} = \begin{bmatrix} 1&1\\ 0&-1 \end{bmatrix}. \]Therefore,
\[ \boxed{AB\neq BA}. \]Practice 10
Matrix products act on vectors from right to left.
If \(A\) acts first,
\[ \mathbf{v}\mapsto A\mathbf{v}. \]If \(B\) acts next,
\[ A\mathbf{v}\mapsto B(A\mathbf{v}). \]Therefore,
\[ \boxed{ B(A\mathbf{v}) = (BA)\mathbf{v} }. \]What to Remember
- Matrix multiplication uses the row-by-column rule.
- The inner dimensions must match for a product to be defined.
- The outer dimensions determine the product's dimensions.
- Matrix-vector multiplication is a special case of matrix multiplication.
- If \(A\) acts first and \(B\) acts second, the combined matrix is \(BA\).
- Matrix multiplication is associative.
- Matrix multiplication is generally not commutative.
- The identity matrix represents the do-nothing transformation.
- An inverse matrix represents an undoing transformation when an inverse exists.
- Repeated transformations are represented by matrix powers.
- A product matrix can replace an entire sequence of linear transformations.
Where We Go Next
Matrix multiplication lets us combine transformation operators and treat an entire sequence as one linear rule. The next step is to recognize that the numerical coordinates of vectors and matrices depend on the basis used to describe the space.
The next lesson, Coordinate Changes and Bases, develops alternative basis vectors, coordinate representations, change-of-basis matrices, and the distinction between a geometric vector and the coordinates used to describe it.