Matrices: order, types and notation

A matrix is a rectangular array of numbers arranged in rows and columns. The matrix

$$A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$$

has $2$ rows and $3$ columns. We call it a $2 \times 3$ matrix (read "2 by 3"). Each number in the matrix is called an entry or element.

Matrices appear everywhere in advanced mathematics — solving systems of equations, transforming geometric shapes, organising data, computer graphics, statistics. For WAEC, the focus is on basic operations: addition, subtraction, scalar multiplication, multiplication, and finding determinants and inverses of $2 \times 2$ matrices.

This lesson introduces matrices and their notation. The next lesson covers matrix operations, determinants, inverses and solving simultaneous equations with matrices.

Learning objectives

By the end of this lesson you should be able to:

• describe a matrix by its rows, columns and order
• identify the element in a given row and column
• recognise row, column, square, zero, identity, diagonal and symmetric matrices
• state when two matrices are equal

Anatomy of a matrix

In the matrix

$$A = \begin{pmatrix} 2 & -3 & 5 \\ 1 & 0 & 7 \\ 4 & 8 & -2 \end{pmatrix}$$

• The order is $3 \times 3$ (3 rows, 3 columns).
• $a_{11} = 2$ (row 1, column 1).
• $a_{12} = -3$ (row 1, column 2).
• $a_{23} = 7$ (row 2, column 3).
• $a_{31} = 4$ (row 3, column 1).
• The element $a_{ij}$ is in the $i$-th row, $j$-th column.

Figure 1. Rows run across, columns run down, and $a_{ij}$ is the element in row $i$, column $j$. WASSCE Tutor diagram.

Reading matrix order

The order is always given as rows × columns (row count first).

• $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ is $2 \times 2$.
• $\begin{pmatrix} 1 & 2 & 3 \end{pmatrix}$ is $1 \times 3$ (a row matrix or row vector).
• $\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$ is $3 \times 1$ (a column matrix or column vector).
• $\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}$ is $3 \times 3$.

Special types of matrices

Row matrix

A matrix with one row: $1 \times n$.

Example: $\begin{pmatrix} 5 & 7 & -2 & 4 \end{pmatrix}$.

Column matrix

A matrix with one column: $m \times 1$.

Example: $\begin{pmatrix} 3 \\ 1 \\ -5 \end{pmatrix}$.

Square matrix

A matrix with equal rows and columns: $n \times n$.

Example: $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ ($2 \times 2$).

Zero matrix (null matrix)

All entries are zero. Denoted $O$ or $\mathbf{0}$.

Example: $O = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$.

Identity matrix

A square matrix with $1$s on the main diagonal and $0$s elsewhere. Denoted $I$.

• $2 \times 2$: $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$.
• $3 \times 3$: $I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$.

The identity matrix plays the role of $1$ for matrix multiplication.

Diagonal matrix

A square matrix where all off-diagonal entries are zero.

Example: $\begin{pmatrix} 5 & 0 & 0 \\ 0 & -3 & 0 \\ 0 & 0 & 2 \end{pmatrix}$.

The identity matrix is a special diagonal matrix.

Triangular matrices

• Upper triangular: all entries below the main diagonal are zero. Example: $\begin{pmatrix} 2 & 3 & 1 \\ 0 & 5 & 4 \\ 0 & 0 & 7 \end{pmatrix}$.
• Lower triangular: all entries above the main diagonal are zero.

Matrix equality

Two matrices are equal if and only if:

1. They have the same order.
2. Every corresponding element is equal.

Example 1

Are these matrices equal?

$A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$.

Same order ($2 \times 2$), same corresponding entries. Yes, equal.

Example 2

Are these equal?

$A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 2 \\ 3 & 5 \end{pmatrix}$.

Different at position $(2, 2)$. Not equal.

Example 3

Are these equal?

$A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 2 & 0 \\ 3 & 4 & 0 \end{pmatrix}$.

Different orders ($2 \times 2$ vs $2 \times 3$). Not equal.

Using matrix equality to solve for unknowns

If two matrices are stated to be equal, set corresponding entries equal.

Example 4

Find $x$ and $y$ if $\begin{pmatrix} x + 1 & 5 \\ 3 & y - 2 \end{pmatrix} = \begin{pmatrix} 4 & 5 \\ 3 & 7 \end{pmatrix}$.

From position $(1, 1)$: $x + 1 = 4$, so $x = 3$. From position $(2, 2)$: $y - 2 = 7$, so $y = 9$.

Example 5

Find $a, b, c$ if $\begin{pmatrix} 2a & 3 \\ 4 & b + 1 \\ c & 0 \end{pmatrix} = \begin{pmatrix} 6 & 3 \\ 4 & 5 \\ -2 & 0 \end{pmatrix}$.

$2a = 6 \to a = 3$. $b + 1 = 5 \to b = 4$. $c = -2$.

Why matrices matter

Matrices appear in many contexts.

Systems of equations

The system $\begin{cases} 2x + 3y = 7 \\ x - y = 1 \end{cases}$ can be written as the matrix equation $\begin{pmatrix} 2 & 3 \\ 1 & -1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 7 \\ 1 \end{pmatrix}$.

Data organisation

A shop's sales for $3$ products across $4$ months: organize as a $3 \times 4$ matrix.

Geometric transformations

Multiplication by a $2 \times 2$ matrix can rotate, scale, or reflect points in the plane. Computer graphics uses this constantly.

Networks

A matrix can represent connections in a network (graph). Entry $a_{ij}$ might be $1$ if vertex $i$ connects to vertex $j$, $0$ otherwise.

Transpose of a matrix

The transpose of $A$, written $A^T$ or $A'$, swaps rows and columns. If $A$ is $m \times n$, then $A^T$ is $n \times m$.

$A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}, \quad A^T = \begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{pmatrix}$.

In the transpose, $a_{ij}^T = a_{ji}$. WAEC may ask basic questions about transpose, but rarely tests it deeply.

A WAEC example

WASSCE examination focus

Practice problems

1. State the order of each matrix. (a) $\begin{pmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{pmatrix}$ (b) $\begin{pmatrix} 7 & -2 & 0 & 3 \end{pmatrix}$ (c) $\begin{pmatrix} 5 \end{pmatrix}$

2. For $A = \begin{pmatrix} 1 & -2 & 5 \\ 0 & 3 & 4 \end{pmatrix}$, find $a_{11}, a_{13}, a_{22}, a_{23}$.

3. Write down the $3 \times 3$ identity matrix.

4. Find $x$ and $y$ if $\begin{pmatrix} x + 2 & 5 \\ 7 & y - 3 \end{pmatrix} = \begin{pmatrix} 8 & 5 \\ 7 & 4 \end{pmatrix}$.

5. Find the transpose of $A = \begin{pmatrix} 1 & 0 & 2 \\ 3 & -1 & 4 \end{pmatrix}$.

6. Is the matrix $\begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix}$ a diagonal matrix? An identity matrix?

7. Find $a, b, c$ if $\begin{pmatrix} 2a + 1 \\ b - 4 \\ 3c \end{pmatrix} = \begin{pmatrix} 7 \\ 2 \\ -9 \end{pmatrix}$.

Worked solutions

1. (a) $3 \times 2$. (b) $1 \times 4$. (c) $1 \times 1$.

2. $a_{11} = 1, a_{13} = 5, a_{22} = 3, a_{23} = 4$.

3. $\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$.

4. $x + 2 = 8 \to x = 6$. $y - 3 = 4 \to y = 7$.

5. $\begin{pmatrix} 1 & 3 \\ 0 & -1 \\ 2 & 4 \end{pmatrix}$.

6. Diagonal matrix: yes (zeros off the diagonal). Identity: no (entries on diagonal are $5$, not $1$).

7. $2a + 1 = 7 \to a = 3$. $b - 4 = 2 \to b = 6$. $3c = -9 \to c = -3$.