A matrix is a rectangular array of numbers enclosed in brackets. The numbers are called elements or entries. Matrices are described by their order: the number of rows × the number of columns.
$$A = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \ a_{31} & a_{32} \end{pmatrix} \quad \text{Order: } 3 \times 2$$
Has exactly one row. Order: \$1 \times n$.
$$\mathbf{r} = \begin{pmatrix} 5 & -2 & 8 \end{pmatrix} \quad \text{Order: } 1 \times 3$$
Has exactly one column. Order: $m \times 1$.
$$\mathbf{c} = \begin{pmatrix} 4 \ 1 \ -3 \end{pmatrix} \quad \text{Order: } 3 \times 1$$
Has equal numbers of rows and columns. Order: $n \times n$.
$$S = \begin{pmatrix} 2 & 7 \ 3 & 1 \end{pmatrix} \quad \text{Order: } 2 \times 2$$
Square matrices have a main diagonal — the entries $a_{11}, a_{22}, \ldots, a_{nn}$.
Every entry is zero. Denoted $\mathbf{0}$ or $O$.
$$O = \begin{pmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \end{pmatrix} \quad \text{Order: } 2 \times 3$$
The zero matrix is the additive identity: $A + O = A$.
A square matrix with 1s on the main diagonal and 0s everywhere else. Denoted $I$ or $I_n$.
$$I_2 = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}, \qquad I_3 = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix}$$
Property: $AI = IA = A$ for any compatible square matrix $A$.
Two matrices are equal if and only if they have the same order and every corresponding entry is equal.
$$\begin{pmatrix} x & 3 \ 1 & y \end{pmatrix} = \begin{pmatrix} 4 & 3 \ 1 & -2 \end{pmatrix} \implies x = 4,\; y = -2$$
A school canteen tracks sales of sandwiches (S), pies (P), and drinks (D) across two days:
| S | P | D | |
|---|---|---|---|
| Mon | 12 | 8 | 25 |
| Tue | 10 | 11 | 30 |
As a matrix: $M = \begin{pmatrix} 12 & 8 & 25 \ 10 & 11 & 30 \end{pmatrix}$, order \$2 \times 3$.
This is neither a row, column, square, zero, nor identity matrix — it is a general rectangular matrix.
EXAM TIP: Always state the order as “rows × columns”. A \$2 \times 3$ matrix has 2 rows and 3 columns — not the other way around.
COMMON MISTAKE: Confusing the zero matrix with a matrix full of zeros that happens to be square. Any-sized matrix of zeros is still called the zero matrix.
