Determinants in a Matrix

The determinant of a square matrix is a special number that gives important information about the matrix — such as whether it has an inverse or whether its transformation squashes space. For WAEC, you’ll mostly work with 2×2 matrices.

 

Determinant of a 2×2 Matrix

Given a 2×2 matrix:

$$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$

The determinant of A is calculated as:

$$\text{det}(A) = ad - bc$$

Example:

$$A = \begin{bmatrix} 3 & 2 \\ 5 & 4 \end{bmatrix}, \quad \text{det}(A) = (3 \times 4) - (2 \times 5) = 12 - 10 = 2$$

 

Determinant of a 3×3 Matrix (Extension/Bonus)

For a 3×3 matrix:

$$A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$$

Use the rule of Sarrus or cofactor expansion (not required in all WAEC questions, but useful for advanced learners).

$$\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$$

Example:

$$\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 1 & 0 & 6 \end{bmatrix}$$

$$\text{det}(A) = 1(4 \times 6 - 5 \times 0) - 2(0 \times 6 - 5 \times 1) + 3(0 \times 0 - 4 \times 1)$$

$$= 1(24 - 0) - 2(0 - 5) + 3(0 - 4) = 24 + 10 - 12 = 22$$

 

Special Determinant Cases

• $\text{det}(A) = 0$: The matrix is called singular (no inverse exists)
• $\text{det}(A) \neq 0$: The matrix is non-singular (an inverse exists)

Example:

$$A = \begin{bmatrix} 4 & 2 \\ 6 & 3 \end{bmatrix}, \quad \text{det}(A) = 4 \times 3 - 2 \times 6 = 12 - 12 = 0$$

This matrix is singular and has no inverse.

 

Worked Example

Practice Problem