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Lesson 32 — Matrix Operations

Sum, scalar multiplication, matrix product. Multiplication as composition of linear transformations.

Used in: 1.º ano EM (álgebra linear elementar) · Equiv. Math I japonês cap. matrizes · Equiv. Klasse 11 alemã (Matrizen)

(AB)_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj}

Choose your door

Rigorous notation, full derivation, hypotheses

Operations

Sum and difference

For matrices of the same dimension: $(A + B)_{ij} = a_{ij} + b_{ij}$

Scalar multiplication

$(\alpha A)_{ij} = \alpha \cdot a_{ij}$

Matrix product

Defined only when the number of columns of $A$ = number of rows of $B$ : $A_{m \times n} \cdot B_{n \times p} = (AB)_{m \times p}$

$\boxed{(AB)_{ij} = \sum_{k=1}^n a_{ik} b_{kj}}$

Properties

Operation	Property
Sum	commutative, associative, identity $O$
Scalar	distributive: $\alpha(A+B) = \alpha A + \alpha B$
Product	associative: $(AB)C = A(BC)$
Product	distributive: $A(B+C) = AB + AC$
Product	NOT commutative in general
Identity	$AI = IA = A$
Zero	$AO = OA = O$

Why the matrix product is "weird"

Because it corresponds to composition of linear transformations: applying first $B$ and then $A$ is the same as applying $AB$ . Order matters because composition matters.

Powers

$A^n = A \cdot A \cdots A$ ( $n$ times), $A^0 = I$ . Defined only for square matrices.

Exercise list

46 exercises · 11 with worked solution (25%)

Application 27Understanding 5Modeling 10Challenge 3Proof 1

Ex. 32.1Application
Compute $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} + \begin{pmatrix} 5 & -1 \\ 0 & 2 \end{pmatrix}$ . (Ans: $\begin{pmatrix} 6 & 1 \\ 3 & 6 \end{pmatrix}$ .)
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Ex. 32.2ApplicationAnswer key
Compute $3 \cdot \begin{pmatrix} 2 & 1 \\ -1 & 0 \end{pmatrix}$ .
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Ex. 32.3Application
Compute $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \cdot \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$ . (Ans: $\begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}$ .)
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Ex. 32.4Application
Compute $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ — what do you get?
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Ex. 32.5ApplicationAnswer key
Compute $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \cdot \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$ .
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Ex. 32.6Application
Multiply a $2 \times 3$ matrix by a $3 \times 2$ — what is the result's dimension? (Ans: $2 \times 2$ .)
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Ex. 32.7Application
Compute $\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} \cdot \begin{pmatrix} 7 \\ 8 \\ 9 \end{pmatrix}$ . (Ans: $\begin{pmatrix} 50 \\ 122 \end{pmatrix}$ .)
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Ex. 32.8Application
Verify $AB \neq BA$ for $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ , $B = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ .
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Ex. 32.9ApplicationAnswer key
$A^2$ for $A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ . (Ans: $I$ .)
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Ex. 32.10Application
Compute $(A + B)^2$ and $(A^2 + 2AB + B^2)$ . When do they coincide? (When $AB = BA$ .)
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Ex. 32.11Application
Compute $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}^2$ .
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Ex. 32.12Application
Multiply $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$ by $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ . (Ans: $(\cos\theta, \sin\theta)^T$ .)
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Ex. 32.13ApplicationAnswer key
Compute the product $\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \cdot \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$ .
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Ex. 32.14Application
Verify distributivity: $A(B+C) = AB + AC$ for $A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$ , $B, C$ of your choice.
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Ex. 32.15Application
For $A_{2 \times 3}$ and $B_{3 \times 4}$ , dimension of $AB$ ? And $BA$ ? ( $BA$ does not exist.)
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Ex. 32.16Application
Compute $\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} \cdot \begin{pmatrix} 4 \\ 5 \end{pmatrix}$ .
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Ex. 32.17ApplicationAnswer key
Show that the product of two diagonal matrices is diagonal — compute explicitly.
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Ex. 32.18Application
Compute $\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}^3$ .
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Ex. 32.19Application
Compute $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}^n$ — find a general formula in terms of $n$ . (Ans: $\begin{pmatrix} 1 & n \\ 0 & 1 \end{pmatrix}$ .)
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Ex. 32.20ApplicationAnswer key
For which $A_{2\times 2}$ does $A^2 = A$ hold? (Idempotent — projection.) Give two examples.
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Ex. 32.21Application
Compute $\begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ -1 & 2 \end{pmatrix}$ .
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Ex. 32.22Application
Compute $\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \begin{pmatrix} 4 & 5 & 6 \end{pmatrix}$ — outer product, dimension $3 \times 3$ .
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Ex. 32.23ApplicationAnswer key
Compute $\begin{pmatrix} 4 & 5 & 6 \end{pmatrix} \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$ — inner product, dimension $1 \times 1$ . (Ans: $32$ .)
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Ex. 32.24Application
Compute $A^4$ for $A = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ . (Ans: $I$ .)
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Ex. 32.25Application
Find $A^2$ for $A = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}$ . And $A^3$ ? (Nilpotent.)
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Ex. 32.26Application
Verify $(AB)^T = B^T A^T$ for $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ , $B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ .
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Ex. 32.27Application
Compute $\begin{pmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{pmatrix} \begin{pmatrix} \cos\beta & -\sin\beta \\ \sin\beta & \cos\beta \end{pmatrix}$ — verify it gives a rotation by $\alpha + \beta$ .
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Ex. 32.28Understanding
Show that multiplying by the identity matrix does not change the matrix. (Direct from the definition.)
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Ex. 32.29Understanding
Show that the zero matrix multiplied gives the zero matrix.
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Ex. 32.30Understanding
Show that if $A$ is diagonal and $B$ is diagonal, $AB$ is diagonal and the diagonal entries multiply.
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Ex. 32.31UnderstandingAnswer key
Show that if $A$ is upper-triangular and $B$ is upper-triangular, $AB$ is upper-triangular.
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Ex. 32.32Understanding
Show that $\text{tr}(AB) = \text{tr}(BA)$ even when $AB \neq BA$ .
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Ex. 32.33Modeling
On a team, players score goals $G$ and assists $A$ . Score: goal $\times 3$ + assist $\times 1$ . Model as a matrix product.
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Ex. 32.34Modeling
In a neural network, layer $\mathbf{y} = W\mathbf{x} + \mathbf{b}$ . For $W \in M_{10 \times 5}$ , what is the dimension of $\mathbf x$ and $\mathbf y$ ?
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Ex. 32.35Modeling
Markov chain: distribution $\pi'$ = $\pi P$ . If $\pi = (0.5, 0.5)$ and $P = \begin{pmatrix} 0.9 & 0.1 \\ 0.2 & 0.8 \end{pmatrix}$ , compute $\pi'$ .
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Ex. 32.36Modeling
Rotation in the plane: $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \cdot \begin{pmatrix} x \\ y \end{pmatrix}$ rotates $(x, y)$ by $\theta$ . Apply it to $(1, 0)$ with $\theta = \pi/2$ .
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Ex. 32.37ModelingAnswer key
PageRank iterates $\mathbf v_{n+1} = M \mathbf v_n$ with $M$ stochastic. What is the eigenvector of $M$ used for?
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Ex. 32.38Modeling
In CG, 2D affine transformation: $T(\mathbf x) = A\mathbf x + \mathbf t$ . How can it be represented via a $3\times 3$ matrix applied to $(x, y, 1)^T$ ?
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Ex. 32.39ModelingAnswer key
In economics, input-output matrix: $\mathbf x = A\mathbf x + \mathbf d$ ⟹ $\mathbf x = (I-A)^{-1}\mathbf d$ . What computational cost?
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Ex. 32.40ModelingAnswer key
In finance, returns $\mathbf r = R \mathbf w$ where $R$ is the matrix of returns by asset × scenario. Model.
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Ex. 32.41Modeling
RGB image $H \times W \times 3$ . Conversion to grayscale: $g = 0.299 r + 0.587 g + 0.114 b$ . How to express it as a matrix product?
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Ex. 32.42Modeling
In control, system $\mathbf x_{k+1} = A\mathbf x_k + B\mathbf u_k$ . After 3 steps, $\mathbf x_3$ as a function of $\mathbf x_0, \mathbf u_0, \mathbf u_1, \mathbf u_2$ .
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Ex. 32.43Challenge
Find $A \neq 0$ and $B \neq 0$ such that $AB = 0$ (matrix zero divisors).
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Ex. 32.44Challenge
Find $A \neq I$ such that $A^2 = I$ (non-trivial involution).
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Ex. 32.45Challenge
Show that if $AB = I$ for square $A, B$ , then $BA = I$ as well (not trivial — depends on finite dimensionality).
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Ex. 32.46Proof
Prove associativity: $(AB)C = A(BC)$ via $\sum_k$ and switching order of sums.

Sources

A First Course in Linear Algebra — Robert A. Beezer · 2022 · EN · GFDL · ch. M and MM. Primary source.
Linear Algebra Done Right — Sheldon Axler · 2024, 4th ed · EN · CC-BY-NC · ch. 3.
Álgebra linear — Wikibooks · alive · PT-BR · CC-BY-SA.