Matrices · 2.2.5
Inverse of a 2×2 matrix
Students learn that the inverse of a square matrix A, written A⁻¹, is the matrix satisfying AA⁻¹ = A⁻¹A = I. For a 2×2 matrix [[a, b], [c, d]], the inverse is found using the formula 1/(ad − bc) × [[d, −b], [−c, a]], provided the determinant ad − bc is not zero.
The official learning standard (2.2.5)
“Explain the meaning of inverse matrix and hence determine the inverse matrix for a 2 × 2 matrix.”
What it means
Students learn that the inverse of a square matrix A, written A⁻¹, is the matrix satisfying AA⁻¹ = A⁻¹A = I. For a 2×2 matrix [[a, b], [c, d]], the inverse is found using the formula 1/(ad − bc) × [[d, −b], [−c, a]], provided the determinant ad − bc is not zero.
How it is examined
Finding a 2×2 inverse matrix is a core Paper 2 skill, usually as the first part of a question that then asks students to solve simultaneous linear equations using that inverse; Paper 1 may test a direct inverse calculation or identifying when an inverse does not exist.
Worked example
Find the inverse of matrix A = [[3, 1], [2, 1]].
- Find the determinant: ad − bc = (3×1) − (1×2) = 3 − 2 = 1
- Since the determinant is 1 (not zero), the inverse exists.
- Swap a and d, negate b and c: [[1, −1], [−2, 3]]
- Divide by the determinant: A⁻¹ = 1/1 × [[1, −1], [−2, 3]] = [[1, −1], [−2, 3]]
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
What happens if the determinant ad − bc equals zero?
The matrix has no inverse; it is called a singular matrix. You cannot divide by zero, so the formula breaks down, and the matrix method cannot be used to solve simultaneous equations built from that matrix.
Why do we swap a and d, and negate b and c?
This rearrangement, together with dividing by the determinant, is exactly what is needed so that A multiplied by this new matrix gives the identity matrix I. It comes from the general inverse formula for 2×2 matrices and should simply be memorised.
How can I check that my inverse matrix is correct?
Multiply your original matrix A by the inverse you found. If the result is the identity matrix [[1, 0], [0, 1]], your inverse is correct.
This quick check is worth doing whenever time allows in the exam.