Matrices
Inverse matrix
The matrix that, multiplied by the original, gives the identity; used to solve simultaneous equations.
| English | Inverse matrix |
|---|---|
| Bahasa Melayu | Matriks songsang |
| 中文 | 逆矩阵 |
How it is used
For A = (2 1; 3 4), determinant = 2×4 − 1×3 = 5, so A⁻¹ = (1/5)(4 −1; −3 2). To solve 2x + y = 4 and 3x + 4y = 6, write (x; y) = A⁻¹(4; 6) = (1/5)(10; 0) = (2; 0), so x = 2, y = 0.
Where it shows up in SPM
The core skill of the Matrices chapter (Form 5). It is a standard Paper 2 question: form the inverse of a 2×2 matrix, then multiply it by a column matrix to solve a pair of simultaneous linear equations.
Don't confuse it with
Frequently asked questions
What is the formula for the inverse of a 2×2 matrix?
For A = (a b; c d), the inverse is (1/(ad − bc))(d −b; −c a). Swap a and d, put minus signs on b and c, then multiply by one over the determinant.
This formula is not given, so memorise it for the exam.
Why does the inverse help solve simultaneous equations?
Writing the equations as A(x; y) = (h; k), multiplying both sides by A⁻¹ leaves (x; y) = A⁻¹(h; k), because A⁻¹A is the identity. So one inverse plus a multiplication gives both unknowns at once, without elimination or substitution.
Can every matrix be inverted?
No. Only a square matrix with a non-zero determinant has an inverse.
If ad − bc = 0, the inverse does not exist because you would be dividing by zero. Non-square matrices, such as a 2×3, have no inverse in the SPM syllabus.