Formula sheet
Inverse of a 2x2 matrix
The inverse formula is printed on the exam formula sheet. You must know how to read a, b, c, d off the matrix in the right positions, compute the determinant ad − bc, swap a and d while negating b and c, and, importantly, that the inverse does not exist when ad − bc = 0.
What the symbols mean
- a, b, c, d the four entries of the matrix A = [[a, b], [c, d]], read row by row
- ad − bc the determinant of A; the inverse exists only when this is not zero
- A⁻¹ the inverse matrix, which multiplies with A to give the identity matrix
Given in the exam, or memorise?
The inverse formula is printed on the exam formula sheet. You must know how to read a, b, c, d off the matrix in the right positions, compute the determinant ad − bc, swap a and d while negating b and c, and, importantly, that the inverse does not exist when ad − bc = 0.
Why it works
The inverse is the matrix that multiplies with A to give the identity matrix I; the formula is the shortcut for the 2×2 case.
- For A = [[a, b], [c, d]], the number ad − bc is called the determinant.
- Swap the leading diagonal (a and d) and negate the other diagonal (b and c) to get the adjoint [[d, −b], [−c, a]].
- Dividing the adjoint by the determinant gives A⁻¹, and multiplying A by it returns the identity matrix.
Worked example 1
Find the inverse of A = [[2, 1], [3, 2]].
- Identify a = 2, b = 1, c = 3, d = 2.
- Determinant ad − bc = (2)(2) − (1)(3) = 4 − 3 = 1.
- A⁻¹ = 1/1 × [[2, −1], [−3, 2]]
- = [[2, −1], [−3, 2]]
Worked example 2
Find the inverse of A = [[3, 4], [1, 2]].
- Identify a = 3, b = 4, c = 1, d = 2.
- Determinant ad − bc = (3)(2) − (4)(1) = 6 − 4 = 2.
- A⁻¹ = 1/2 × [[2, −4], [−1, 3]]
- = [[1, −2], [−½, 3/2]]
Where students go wrong
- Forgetting to swap a and d, the leading diagonal entries change places before anything else.
- Negating the wrong entries, you negate b and c (the off-diagonal), not a and d.
- Computing the determinant as ad + bc or bc − ad instead of ad − bc.
- Trying to invert when ad − bc = 0, a singular matrix has no inverse (you would be dividing by zero).
Use it with
Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)
Frequently asked questions
What happens if the determinant is zero?
The matrix has no inverse. Since the formula divides by ad − bc, a determinant of 0 means dividing by zero, which is impossible.
Such a matrix is called singular. In an exam, getting ad − bc = 0 is a signal to state that the inverse does not exist.
Do I have to divide every entry by the determinant?
Yes. The 1/(ad − bc) multiplies the whole matrix, so every one of the four entries is divided by the determinant.
It is neat to leave the fraction outside the matrix, like ½[[2, −4], [−1, 3]], but each entry inside is still scaled by it.
How can I check my inverse is correct?
Multiply your answer by the original matrix. If A⁻¹ is right, A × A⁻¹ gives the identity matrix [[1, 0], [0, 1]].
This quick check catches sign slips and swapped entries, and it works for any 2×2 inverse you compute.