Matrices
1.5 Matrix Inversion
Definition of the matrix inverse, inverse of 2×2 via formula, inverse of n×n via adjoint matrix, singularity condition, and laws of matrix algebra.
1.5 Matrix Inversion
Definition 1.24: If AB = I and BA = I, then B is the inverse of A, written A−1. The inverse exists only if A is square and non-singular.
Inverse of a 2×2 Matrix
A = [a,b; c,d] → A−1 = 1det(A) · [d,−b; −c,a]
det(A) = ad − bc must be ≠ 0. If det(A) = 0, A is singular and has no inverse.
Example 1.15: A = [[8,2],[4,3]]
det(A) = 24−8 = 16
→
A−1 = 116 ×
| 3 | −2 |
| −4 | 8 |
Inverse of n×n Matrix via Adjoint
A−1 = adj(A)det(A) where adj(A) = CT (transpose of cofactor matrix)
Example 1.16: A = [[1,2,3],[7,5,6],[4,8,9]]
Step 1: Cofactor matrix C
| −3 | −39 | 36 |
| 6 | −3 | 0 |
| −3 | 15 | −9 |
Step 2: adj(A) = CT
| −3 | 6 | −3 |
| −39 | −3 | 15 |
| 36 | 0 | −9 |
Step 3: det(A) = 1(−3) − 2(63−24) + 3(56−20) = −3 − 78 + 108 = 27
Step 4: A−1 = adj(A)27
| −19 | 29 | −19 |
| −139 | −19 | 59 |
| 43 | 0 | −13 |
1.6 Laws of Matrix Algebra
- If AB = I then B = A−1 and A = B−1
- (A−1)−1 = A
- (kA)−1 = k−1A−1
- (AT)−1 = (A−1)T
- det(A−1) = 1det(A)
- (AB)−1 = B−1A−1 (reverse order!)
Warning: (AB)−1 = B−1A−1 NOT A−1B−1 — order is reversed!
Example 1.17: If BACD = I, express A in terms of inverses.
B−1BACD = ACD = B−1 → AC = B−1D−1 → A = B−1D−1C−1