How to Solve Matrices Questions in ECZ Mathematics

Mathematics (Grade 12 & GCE)  |  Updated March 2026  |  12 min read

Matrices appear consistently in ECZ Grade 12 and GCE Mathematics Paper 1 (short 2-mark or 3-mark questions) and Paper 2 (Section A 5-mark matrix equations). Mastering determinant and inverse matrix calculations guarantees easy marks.

Senior Secondary Tip

Order of a matrix is always stated as Rows $\times$ Columns ($m \times n$). Think of "RC Cola" (Rows first, then Columns)!

1. Basic Matrix Operations

  • Matrix Addition & Subtraction: Matrices must be of the EXACT same order. Add or subtract corresponding entry positions.
  • Scalar Multiplication: Multiply every single entry inside the matrix by the constant number outside.
  • Matrix Multiplication: Use the Row-by-Column dot product method.

2. Determinant and Inverse of a $2 \times 2$ Matrix

For a general $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$:

\[ \text{Determinant } |A| = ad - bc \]

The inverse matrix $A^{-1}$ exists if $|A| \ne 0$:

\[ A^{-1} = \frac{1}{|A|} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]

Worked Example: Finding Inverse and Solving Equations

Problem: Given matrix $A = \begin{pmatrix} 3 & 2 \\ 5 & 4 \end{pmatrix}$:

  1. Find the determinant $|A|$.
  2. Find the inverse matrix $A^{-1}$.
  3. Hence, solve the simultaneous equations: \[ 3x + 2y = 8 \] \[ 5x + 4y = 14 \]

Solution Part (i): Determinant

\[ |A| = (3 \times 4) - (2 \times 5) = 12 - 10 = 2 \]

Solution Part (ii): Inverse Matrix

\[ A^{-1} = \frac{1}{2} \begin{pmatrix} 4 & -2 \\ -5 & 3 \end{pmatrix} = \begin{pmatrix} 2 & -1 \\ -2.5 & 1.5 \end{pmatrix} \]

Solution Part (iii): Solving Matrix Equation $\begin{pmatrix} x \\ y \end{pmatrix} = A^{-1} \begin{pmatrix} 8 \\ 14 \end{pmatrix}$

\[ \begin{pmatrix} x \\ y \end{pmatrix} = \frac{1}{2} \begin{pmatrix} 4 & -2 \\ -5 & 3 \end{pmatrix} \begin{pmatrix} 8 \\ 14 \end{pmatrix} \] \[ \begin{pmatrix} 4(8) + (-2)(14) \\ -5(8) + 3(14) \end{pmatrix} = \begin{pmatrix} 32 - 28 \\ -40 + 42 \end{pmatrix} = \begin{pmatrix} 4 \\ 2 \end{pmatrix} \] \[ \begin{pmatrix} x \\ y \end{pmatrix} = \frac{1}{2} \begin{pmatrix} 4 \\ 2 \end{pmatrix} = \begin{pmatrix} 2 \\ 1 \end{pmatrix} \]

Thus, $x = 2$ and $y = 1$.

Singular Matrix Exam Trick

If an ECZ question states: "Find the value of $k$ for which matrix $\begin{pmatrix} k & 4 \\ 3 & 6 \end{pmatrix}$ is singular", set determinant to ZERO: $(k \times 6) - (4 \times 3) = 0 \implies 6k = 12 \implies k = 2$!

Teacher Strategy

Have learners swap main diagonal elements ($a$ and $d$) and change signs of off-diagonal elements ($b$ and $c$) in red ink on flashcards to memorize $A^{-1}$ steps easily.

Frequently Asked Questions

Q1: What condition must be met to multiply two matrices A and B?

Matrix multiplication AB is possible only when the number of COLUMNS in matrix A equals the number of ROWS in matrix B. If A is of order m x n and B is n x p, the product AB is of order m x p.

Q2: How do you calculate the determinant and inverse of a 2x2 matrix?

For a 2x2 matrix A = [[a, b], [c, d]], the determinant is |A| = ad - bc. The inverse is A^-1 = (1 / |A|) * [[d, -b], [-c, a]], provided |A| ≠ 0.

Q3: What is a singular matrix?

A singular matrix is a matrix whose determinant is equal to ZERO (|A| = 0). Singular matrices do NOT have an inverse.

Ace ECZ Mathematics Matrices Questions

Download past matrix questions with step-by-step worked solutions for Grade 12 and GCE candidates.

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