Matrices

⚡ Key Takeaways

A matrix is a rectangular array of numbers arranged in rows and columns, written $A = [a_{ij}]$ . Matrices encode and manipulate systems of linear equations, geometric transformations, and data tables. They are added entrywise but multiplied by a special row-times-column rule that is not commutative. Used everywhere from computer graphics to economics; the modern theory was crystallized by Arthur Cayley in 1858.

💡 The Core Insight

A matrix is simultaneously two things: a static table of data and a dynamic operation. The single rule of matrix multiplication — "each entry is a row dotted with a column" — is the mechanism that lets a matrix act on vectors, compose transformations, and solve equations all at once. Master that one rule and most of the subject follows.

DefinitionMatrix and Basic Notation

An $m \times n$ matrix $A$ over the real numbers is an array

1

\quad A = \begin{bmatrix} 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{bmatrix}

Derivation hint: this is a definition — note $a_{ij}$ sits in row $i$ , column $j$ ; the row index always comes first. Here $m$ is the number of rows, $n$ the number of columns, and $a_{ij} \in \mathbb{R}$ is the entry in row $i$ , column $j$ . The pair $(m,n)$ is the order or dimension. A matrix with $m=n$ is square. A $1 \times n$ matrix is a row vector; an $m \times 1$ matrix is a column vector.

DefinitionAddition and Scalar Multiplication

For two $m \times n$ matrices $A = [a_{ij}]$ and $B = [b_{ij}]$ , and a scalar $\lambda \in \mathbb{R}$ :

2

\quad (A+B)_{ij} = a_{ij} + b_{ij}, \qquad (\lambda A)_{ij} = \lambda\, a_{ij}

Derivation hint: addition and scalar multiplication act entrywise; both matrices must have identical order to be added.

DefinitionMatrix Multiplication

If $A$ is $m \times n$ and $B$ is $n \times p$ , then the product $C = AB$ is the $m \times p$ matrix with entries

3

\quad c_{ij} = \sum_{k=1}^{n} a_{ik}\, b_{kj}

Derivation hint: the $(i,j)$ entry is the dot product of row $i$ of $A$ with column $j$ of $B$ ; the "inner" dimensions $n$ must match. The number of columns of $A$ must equal the number of rows of $B$ for the product to be defined.

DefinitionIdentity and Inverse

The $n \times n$ identity matrix $I_n$ has 1's on the main diagonal and 0's elsewhere: $(I_n)_{ij} = \delta_{ij}$ (the Kronecker delta). A square matrix $A$ is invertible (nonsingular) if there exists $A^{-1}$ with

4

\quad A A^{-1} = A^{-1} A = I_n

Derivation hint: the inverse undoes $A$ ; it exists precisely when $\det A \neq 0$ (see below).

TheoremDeterminant of a 2×2 Matrix and Inverse Criterion

For $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ , the determinant is

5

\quad \det A = ad - bc

Derivation hint: it is the signed area of the parallelogram spanned by the columns $(a,c)$ and $(b,d)$ . $A$ is invertible if and only if $\det A \neq 0$ , and then

6

\quad A^{-1} = \frac{1}{ad-bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}

Derivation hint: verify $AA^{-1}=I$ directly by multiplying; the swap of $a,d$ and sign flips on $b,c$ come from solving $AX=I$ .

ProofWhy the 2×2 Inverse Formula Works

We seek $X = \begin{bmatrix} x & y \\ z & w \end{bmatrix}$ with $AX = I_2$ .

Step 1. Write out $AX = I$ using rule [3]:

7

\quad \begin{bmatrix} ax+bz & ay+bw \\ cx+dz & cy+dw \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

Derivation hint: multiply row-by-column and set equal to the identity entry by entry.

Step 2. This gives two independent $2\times 2$ systems:

8

\quad ax+bz=1,\;\; cx+dz=0 \quad\text{and}\quad ay+bw=0,\;\; cy+dw=1

Derivation hint: the first column of $I$ determines $x,z$ ; the second column determines $y,w$ .

Step 3. Solve the first system by elimination. From $cx+dz=0$ we get $z = -cx/d$ (assuming $d\neq 0$ ); substituting and clearing gives $x = d/(ad-bc)$ and $z = -c/(ad-bc)$ .

Step 4. Solving the second system identically yields $y = -b/(ad-bc)$ , $w = a/(ad-bc)$ . Assembling $X$ gives formula [6]. The division by $ad-bc$ shows the inverse exists exactly when $\det A \neq 0$ . $\blacksquare$

∎

mathematician · Arthur Cayley | 1821–1895

The English mathematician Arthur Cayley founded the abstract theory of matrices in his 1858 Memoir on the Theory of Matrices, where he first treated a matrix as a single algebraic object with its own arithmetic — defining addition, multiplication, and inverses. He recognized that matrix multiplication is non-commutative and stated (verifying for $2\times2$ and $3\times3$ ) what is now the Cayley–Hamilton theorem: every square matrix satisfies its own characteristic equation. A lawyer for fourteen years who published over 200 papers in his spare time before taking a Cambridge chair, Cayley also developed invariant theory and contributed to group theory and geometry.

🧠 Intuition

Think of a matrix as a machine that transforms vectors. Feed in a column vector $\mathbf{x}$ , and $A\mathbf{x}$ comes out as a new vector — possibly rotated, stretched, sheared, or projected. The columns of $A$ tell you exactly where the basis vectors land: column $j$ is the image of the $j$ -th unit vector. Multiplying two matrices $AB$ means "do $B$ first, then $A$ " — composing two transformations into one. This is why order matters: rotating then reflecting is generally different from reflecting then rotating. Addition, by contrast, is gentle and commutative because it just superimposes two tables entry by entry.

Figure

A matrix as a transformation: the unit square mapped by a shear matrix

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ExampleMultiplying Two Matrices

Given: $A = \begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 4 \\ 5 & 2 \end{bmatrix}$ . Compute $AB$ .

Step 1 — check dimensions. $A$ is $2\times2$ , $B$ is $2\times2$ ; inner dimensions match, so $AB$ is $2\times2$ .

Step 2 — apply rule [3], each entry = row · column.

$(AB)_{11} = (2)(1)+(1)(5) = 2+5 = 7$
$(AB)_{12} = (2)(4)+(1)(2) = 8+2 = 10$
$(AB)_{21} = (0)(1)+(3)(5) = 0+15 = 15$
$(AB)_{22} = (0)(4)+(3)(2) = 0+6 = 6$

Step 3 — assemble.

\boxed{AB = \begin{bmatrix} 7 & 10 \\ 15 & 6 \end{bmatrix}}

(Check non-commutativity: computing $BA$ gives $\begin{bmatrix} 2 & 13 \\ 10 & 11 \end{bmatrix} \neq AB$ .)

ExampleInverting a Matrix and Solving a System

Given: Solve $\begin{cases} 3x + 2y = 8 \\ x + 4y = 6 \end{cases}$ using a matrix inverse.

Step 1 — write as $A\mathbf{x}=\mathbf{b}$ . $A = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}$ , $\mathbf{b} = \begin{bmatrix} 8 \\ 6 \end{bmatrix}$ .

Step 2 — determinant (eq [5]). $\det A = (3)(4)-(2)(1) = 12-2 = 10 \neq 0$ , so $A^{-1}$ exists.

Step 3 — inverse (eq [6]).

A^{-1} = \frac{1}{10}\begin{bmatrix} 4 & -2 \\ -1 & 3 \end{bmatrix}

Step 4 — multiply $\mathbf{x} = A^{-1}\mathbf{b}$ .

\mathbf{x} = \frac{1}{10}\begin{bmatrix} 4 & -2 \\ -1 & 3 \end{bmatrix}\begin{bmatrix} 8 \\ 6 \end{bmatrix} = \frac{1}{10}\begin{bmatrix} 32-12 \\ -8+18 \end{bmatrix} = \frac{1}{10}\begin{bmatrix} 20 \\ 10 \end{bmatrix}

\boxed{x = 2, \quad y = 1}

(Check: $3(2)+2(1)=8$ ✓ and $1(2)+4(1)=6$ ✓.)

⚠️ Common mistake

Matrix multiplication is NOT commutative. In general $AB \neq BA$ , and sometimes only one product is even defined (a $2\times3$ times a $3\times4$ works one way only). Never "cancel" or reorder factors freely. Likewise, there is no matrix division — you multiply by an inverse, and you must be careful whether it goes on the left ( $A^{-1}B$ ) or the right ( $BA^{-1}$ ), since these differ.

Counterexample

The familiar fact "if $xy = 0$ then $x=0$ or $y=0$ " fails for matrices. Take

A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \qquad B = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}.

Then $AB = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$ , the zero matrix, even though neither $A$ nor $B$ is zero. These are zero divisors; both have determinant $0$ , so neither is invertible — which is exactly why the cancellation law breaks.

connection

Matrices are the computational engine behind nearly all of Linear Algebra. Each matrix represents a Linear Transformations once bases are chosen, with its columns giving the images of basis vectors. The Determinant measures how a matrix scales area/volume and detects invertibility. Solving $A\mathbf{x}=\mathbf{b}$ is the matrix form of Systems of Linear Equations, usually carried out by Gaussian Elimination rather than computing inverses. The deeper structure of a square matrix is captured by its Eigenvalues and Eigenvectors, which reveal directions that the matrix merely stretches. Matrices also realize the abstract notion of a Vector Spaces of linear maps under addition and composition.

🔬 Where this appears

⏳ History

- c. 200 BCE: The Chinese text Nine Chapters on the Mathematical Art solves linear systems with a method essentially equivalent to Gaussian elimination on a numerical array.

- 1683: Seki Takakazu in Japan and, independently, 1693 Gottfried Leibniz in Europe introduce determinants to analyze systems of equations.

- 1750: Gabriel Cramer publishes Cramer's Rule, expressing solutions of linear systems via determinants.

- 1812: Augustin-Louis Cauchy proves the multiplicative property

\det(AB) = \det A \cdot \det B

and systematizes determinant theory.

- 1844–1850: James Joseph Sylvester coins the word "matrix" (Latin for "womb"), viewing it as the array from which determinants (minors) are born.

- 1858: Arthur Cayley's Memoir on the Theory of Matrices establishes matrix algebra as an independent subject and states the Cayley–Hamilton theorem.

- 1878: Ferdinand Georg Frobenius develops the theory of canonical forms, rank, and the characteristic polynomial rigorously.

- 1925: Werner Heisenberg, Max Born, and Pascual Jordan formulate matrix mechanics, the first complete version of quantum theory, founded on non-commuting matrices.

- 1947–present: John von Neumann, Alan Turing, and others develop numerical linear algebra (LU decomposition, condition numbers), making matrices the workhorse of digital computation.

open-problem

Computing matrix products fast is a surprisingly deep open question. The schoolbook rule [3] costs about $n^3$ multiplications for $n\times n$ matrices. In 1969 Volker Strassen shocked everyone by doing it in roughly $n^{2.807}$ operations, and successive refinements (Coppersmith–Winograd and beyond) have pushed the exponent below $2.372$ . But the true minimal exponent $\omega$ — the smallest number such that multiplication is possible in $n^{\omega+\varepsilon}$ time — is unknown; many conjecture $\omega = 2$ , which would be essentially optimal, but no proof exists. Separately, the numerical stability of inversion is subtle: nearly-singular matrices (small but nonzero determinant) amplify rounding errors dramatically, quantified by the condition number, so in practice one almost never forms $A^{-1}$ explicitly.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also