Matrices — chapter notes

A matrix is a rectangular arrangement of numbers in rows and columns. A matrix with $m$ rows and $n$ columns has order $m \times n$ and $mn$ elements. The element in row $i$, column $j$ is $a_{ij}$; we write $A = [a_{ij}]_{m \times n}$.

Matrices organise information. The stock of two shop branches across three items is a $2 \times 3$ matrix, and each entry's position tells you its meaning.

From a rule: to construct $A$ with $a_{ij} = 3i - j$ ($2 \times 3$), fix the row and run through the columns: $A = \begin{bmatrix} 2 & 1 & 0 \\ 5 & 4 & 3 \end{bmatrix}$.

• Row matrix: one row ($1 \times n$). Column matrix: one column ($m \times 1$).
• Square matrix: $m = n$. Its main diagonal is $a_{11}, a_{22}, \dots, a_{nn}$.
• Diagonal matrix: square, with $a_{ij} = 0$ whenever $i \ne j$.
• Scalar matrix: diagonal with all diagonal entries equal to some $k$; it equals $kI$.
• Identity matrix $I_n$: diagonal entries $1$, others $0$.
• Zero matrix $O$: every entry $0$ (any order).

Every identity matrix is scalar, and every scalar matrix is diagonal.

Equality: $A = B$ when they have the same order and $a_{ij} = b_{ij}$ for every $i$, $j$. One matrix equation gives one equation per position, e.g. $\begin{bmatrix} 2a & b + 1 \\ c - 2 & 4 \end{bmatrix} = \begin{bmatrix} 6 & -3 \\ 7 & d \end{bmatrix}$ gives $a = 3$, $b = -4$, $c = 9$, $d = 4$.

Addition (same order only): $A + B = [a_{ij} + b_{ij}]$. Scalar multiple: $kA = [k a_{ij}]$. Difference: $A - B = A + (-1)B$.

Properties: $A + B = B + A$; $(A + B) + C = A + (B + C)$; $A + O = A$; $A + (-A) = O$; $k(A + B) = kA + kB$; $(k + l)A = kA + lA$.

If $A$ is $m \times n$ and $B$ is $n \times p$, then $AB$ is the $m \times p$ matrix with $c_{ik} = a_{i1}b_{1k} + a_{i2}b_{2k} + \dots + a_{in}b_{nk},$ i.e. row $i$ of $A$ times column $k$ of $B$. If the inner numbers differ, $AB$ is not defined.

Laws that hold: $(AB)C = A(BC)$; $A(B + C) = AB + AC$; $(A + B)C = AC + BC$; $AI = IA = A$ for square $A$.

Laws that fail: $AB \ne BA$ in general (e.g. $A = \begin{bmatrix} 1 & 1 \\ 0 & 2 \end{bmatrix}$, $B = \begin{bmatrix} 3 & 0 \\ 1 & 1 \end{bmatrix}$); $AB = O$ is possible with $A \ne O$, $B \ne O$; $(A + B)^2 = A^2 + AB + BA + B^2$.

Matrix polynomials: for $f(x) = x^2 - 5x + 7$, $f(A) = A^2 - 5A + 7I$. For $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$ this is $O$.

A $2 \times 2$ matrix times a column $\begin{bmatrix} x \\ y \end{bmatrix}$ moves the point $(x, y)$: the picture shows the unit square under $\begin{bmatrix} 2 & 1 \\ 0 & 1 \end{bmatrix}$.

The transpose of $A = [a_{ij}]_{m \times n}$ is $A^T = [a_{ji}]_{n \times m}$: rows of $A$ become columns of $A^T$.

Properties: $(A^T)^T = A$; $(kA)^T = kA^T$; $(A + B)^T = A^T + B^T$; $(AB)^T = B^TA^T$ (reversal law).

Example: for $A = \begin{bmatrix} 1 & 2 \\ 3 & 0 \end{bmatrix}$, $B = \begin{bmatrix} 2 & -1 \\ 1 & 4 \end{bmatrix}$, both $(AB)^T$ and $B^TA^T$ equal $\begin{bmatrix} 4 & 6 \\ 7 & -3 \end{bmatrix}$.

A square matrix is symmetric if $A^T = A$ ($a_{ji} = a_{ij}$) and skew-symmetric if $A^T = -A$ ($a_{ji} = -a_{ij}$). A skew-symmetric matrix has every diagonal entry $0$. Only $O$ is both.

Theorem. For any square $A$, $A + A^T$ is symmetric and $A - A^T$ is skew-symmetric, so $A = \underbrace{\tfrac12(A + A^T)}_{P\ \text{symmetric}} + \underbrace{\tfrac12(A - A^T)}_{Q\ \text{skew-symmetric}}.$

Example: $\begin{bmatrix} 1 & 6 \\ 2 & 5 \end{bmatrix} = \begin{bmatrix} 1 & 4 \\ 4 & 5 \end{bmatrix} + \begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix}$.

A square matrix $A$ of order $n$ is invertible if there is a square matrix $B$ of order $n$ with $AB = BA = I$; $B$ is the inverse $A^{-1}$. Non-square matrices are never invertible, and some square matrices (such as $O$, or $\begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}$) are not.

Uniqueness. If $B$ and $C$ are inverses of $A$: $B = BI = B(AC) = (BA)C = IC = C$.

Finding or checking an inverse (this chapter):

• Verify a given $B$: show $AB = I$ and $BA = I$.
• Solve $AB = I$ for unknown entries $p, q, r, s$: two pairs of linear equations, e.g. $\begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix}^{-1} = \begin{bmatrix} 3 & -5 \\ -1 & 2 \end{bmatrix}$.
• From a relation: if $A^2 - 5A + 7I = O$ then $A(5I - A) = 7I$, so $A^{-1} = \frac17(5I - A)$.

(The adjoint method comes in the chapter on determinants; elementary row operations are not in the 2026–27 syllabus.)