Matrix inverse and equations solver
Find the inverse of a matrix by the adjoint method - the determinant, minors, cofactors and adjoint shown - or solve simultaneous linear equations by the matrix method, X = A⁻¹B, as ISC Mathematics asks.
Answer
- |A|
- -11
A⁻¹ =
| 4/11 | 3/11 |
| 1/11 | -2/11 |
Determinant
|A| = (2)(-4) - (3)(1) = -11
Minors and cofactors
Minors =
| -4 | 1 |
| 3 | 2 |
Cofactors =
| -4 | -1 |
| -3 | 2 |
Cofactor Cᵢⱼ = (-1)ⁱ⁺ʲ × minor Mᵢⱼ.
Adjoint and inverse
- adj A is the transpose of the cofactor matrix:adj A =
-4 -3 -1 2 - A⁻¹ = (1/|A|) adj A = (-1/11) adj A:A⁻¹ =
4/11 3/11 1/11 -2/11 - Check: A·A⁻¹ = I ✓
Transpose
Aᵀ =
| 2 | 1 |
| 3 | -4 |
A is neither symmetric nor skew-symmetric.
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