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.
Has exactly one row. Order: \(1 \times n\).
Has exactly one column. Order: \(m \times 1\).
Has equal numbers of rows and columns. Order: \(n \times n\).
Square matrices have a main diagonal — the entries \(a_{11}, a_{22}, \ldots, a_{nn}\).
Every entry is zero. Denoted \(\mathbf{0}\) or \(O\).
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 \(I_n\).
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.
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.
