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Lesson 31 — Introduction to Matrices

Matrix as a rectangular table of numbers. Notation, dimensions, equality, special types.

Used in: 1.º ano EM (15 anos) · Equiv. Math II japonês · Equiv. Klasse 10 alemã · Pré-cálculo norte-americano §11.5

A = (a_{ij})_{m \times n} = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}

Choose your door

Rigorous notation, full derivation, hypotheses

Definition and types

Special types

Type	Definition
Square	$m = n$
Row	$1 \times n$
Column	$m \times 1$
Diagonal	square with $a_{ij} = 0$ for $i \neq j$
Identity $I_n$	diagonal with $a_{ii} = 1$
Zero $O$	all entries zero
Upper triangular	$a_{ij} = 0$ for $i > j$
Lower triangular	$a_{ij} = 0$ for $i < j$
Symmetric	$A^T = A$ , that is $a_{ij} = a_{ji}$
Skew-symmetric	$A^T = -A$ , that is $a_{ij} = -a_{ji}$
Scalar	diagonal with $a_{ii} = k$ constant

Diagonal of a square matrix

The main diagonal is $\{a_{ii}\}$ . Trace: $\text{tr}(A) = \sum_{i=1}^n a_{ii}$ .

Construction rule

Often $A$ is defined via a formula $a_{ij} = f(i, j)$ . Examples:

$a_{ij} = i + j$
$a_{ij} = (-1)^{i+j}$
$a_{ij} = \delta_{ij}$ (Kronecker delta — produces the identity)

Exercise list

46 exercises · 11 with worked solution (25%)

Application 26Understanding 6Modeling 10Challenge 3Proof 1

Ex. 31.1Application
Identify the dimension of $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{pmatrix}$ . (Ans: $3 \times 2$ .)
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Ex. 31.2Application
Write a $3 \times 3$ identity matrix.
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Ex. 31.3Application
Write a $2 \times 4$ zero matrix.
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Ex. 31.4Application
For $A = \begin{pmatrix} 5 & -2 \\ 3 & 4 \end{pmatrix}$ , identify $a_{11}, a_{12}, a_{21}, a_{22}$ . (Ans: $5, -2, 3, 4$ .)
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Ex. 31.5ApplicationAnswer key
Construct $A_{2 \times 3}$ such that $a_{ij} = i + j$ .
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Ex. 31.6ApplicationAnswer key
Construct $A_{3 \times 3}$ such that $a_{ij} = i \cdot j$ .
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Ex. 31.7ApplicationAnswer key
Check whether $A = \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix}$ is symmetric. (Ans: yes.)
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Ex. 31.8Application
Check whether $A = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ is skew-symmetric.
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Ex. 31.9ApplicationAnswer key
Trace of $\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}$ . (Ans: $15$ .)
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Ex. 31.10Application
For which $x$ is $\begin{pmatrix} x & 2 \\ 3 & x \end{pmatrix} = \begin{pmatrix} 5 & 2 \\ 3 & 5 \end{pmatrix}$ ?
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Ex. 31.11ApplicationAnswer key
Construct any $3 \times 3$ upper-triangular matrix.
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Ex. 31.12Application
Construct a $3 \times 3$ diagonal matrix with diagonal entries $2, -1, 5$ . Compute the trace. (Ans: $6$ .)
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Ex. 31.13Application
Identify entry $a_{32}$ of $\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}$ . (Ans: $8$ .)
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Ex. 31.14Application
For $A_{m \times n}$ with $m = n$ , what class is the matrix?
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Ex. 31.15ApplicationAnswer key
How many entries does a $4 \times 5$ matrix have? (Ans: $20$ .)
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Ex. 31.16Application
Construct $A_{2 \times 2}$ with $a_{ij} = (-1)^{i+j}$ .
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Ex. 31.17Application
Verify whether $\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ is the identity.
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Ex. 31.18ApplicationAnswer key
Construct $A_{3 \times 3}$ with $a_{ij} = \max(i, j)$ .
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Ex. 31.19Application
Decide: is the matrix $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ symmetric?
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Ex. 31.20ApplicationAnswer key
Construct a $5 \times 5$ identity matrix. How many zeros does it have? (Ans: $20$ .)
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Ex. 31.21Application
Construct $A_{3 \times 3}$ with $a_{ij} = |i - j|$ . Is it symmetric?
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Ex. 31.22Application
Construct $A_{4 \times 4}$ with $a_{ij} = \delta_{ij}$ (Kronecker delta). What matrix is this?
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Ex. 31.23Application
How many non-zero entries does $I_n$ have? (Ans: $n$ .)
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Ex. 31.24Application
Construct a $4 \times 4$ lower-triangular matrix with $a_{ij} = i + j$ if $i \geq j$ .
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Ex. 31.25Application
Identify whether $\begin{pmatrix} 2 & -1 & 3 \\ -1 & 5 & 0 \\ 3 & 0 & 7 \end{pmatrix}$ is symmetric.
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Ex. 31.26Application
Find $x, y$ so that $\begin{pmatrix} 2x & 3 \\ 5 & y+1 \end{pmatrix} = \begin{pmatrix} 8 & 3 \\ 5 & 7 \end{pmatrix}$ . (Ans: $x = 4, y = 6$ .)
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Ex. 31.27Understanding
Show that a symmetric matrix must be square.
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Ex. 31.28UnderstandingAnswer key
Show that a skew-symmetric matrix has zero diagonal.
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Ex. 31.29Understanding
Show that if $A$ is symmetric, then $a_{ij} = a_{ji}$ for all $i, j$ .
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Ex. 31.30Understanding
How many $3 \times 3$ symmetric matrices exist with entries in $\{0, 1\}$ ? (Ans: $2^6 = 64$ — 6 independent entries are chosen.)
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Ex. 31.31Understanding
Show that any square matrix can be written as the sum of a symmetric + skew-symmetric matrix: $A = \frac{A + A^T}{2} + \frac{A - A^T}{2}$ .
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Ex. 31.32Understanding
Verify the decomposition above for $A = \begin{pmatrix} 1 & 4 \\ 2 & 5 \end{pmatrix}$ .
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Ex. 31.33ModelingAnswer key
Grades of 3 students in 4 subjects: build a $3 \times 4$ matrix with made-up data.
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Ex. 31.34ModelingAnswer key
Distances between 4 cities: $4 \times 4$ symmetric matrix with zero diagonal.
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Ex. 31.35Modeling
$2 \times 3$ grayscale image. Each entry from 0 (black) to 255 (white). Construct an example.
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Ex. 31.36Modeling
Price table by store × product: build a $3 \times 4$ matrix (3 stores, 4 products).
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Ex. 31.37Modeling
In ML, dataset with $n$ samples × $d$ features: what is the matrix dimension?
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Ex. 31.38Modeling
Linear system $\begin{cases} 2x + 3y = 5 \\ x - y = 1 \end{cases}$ — write the coefficient matrix and the augmented matrix.
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Ex. 31.39Modeling
Adjacency matrix of a 4-vertex graph with edges $\{1\text{-}2, 2\text{-}3, 3\text{-}4, 1\text{-}4\}$ .
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Ex. 31.40Modeling
In finance, $5 \times 5$ correlation matrix among stocks: symmetric, diagonal $= 1$ . How many unique values? (Ans: $10$ .)
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Ex. 31.41Modeling
In production, cost $\times$ quantity matrix: each entry is the total cost of that combination.
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Ex. 31.42Modeling
In control, state $\mathbf x \in \mathbb R^3$ with dynamics matrix $A \in M_{3\times 3}$ . How many entries?
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Ex. 31.43Challenge
Show that the dimension of the space of $n \times n$ symmetric matrices is $n(n+1)/2$ .
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Ex. 31.44Challenge
Show that the dimension of the space of $n \times n$ skew-symmetric matrices is $n(n-1)/2$ .
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Ex. 31.45Challenge
Construct $A_{3 \times 3}$ such that $a_{ij} = \binom{i+j-2}{j-1}$ . Recognize the pattern? (Pascal rows — shifted Hilbert matrix.)
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Ex. 31.46Proof
Prove that if $A$ is simultaneously symmetric and skew-symmetric, then $A = O$ .

Sources

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