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Question
Find the inverse of the matrix A by using adjoint method.
where A = `[(-3, -1, 1),(0, 0, 1),(-15, 6, -6)]`
Solution
A = `[(-3, -1, 1),(0, 0, 1),(-15, 6, -6)]`
∴ |A| = – 3(0 – 6) + 1(0 + 15) + 1(0 – 0)
= 18 + 15
= 33 ≠ 0
∴ A–1 exists
To find cofactors.
m11 = 0 – 6 = – 6
m12 = 0 + 15 = 15
m13 = 0 + 0 = 0
m21 = 6 – 6 = 0
m22 = 18 + 15 = 33
m23 = – 18 – 15 = – 33
m31 = – 1 – 0 = – 1
m32 = – 3 – 0 = – 3
m33 = 0 – 0 = 0
A11 = (– 1)2 (– 6) = – 6
A12 = (– 1)3.15 = – 15
A13 = (– 1)4.0 = 0
A21 = (– 1)3.0 = 0
A22 = (– 1)4.33 = 33
A23 = (– 1)5. (– 33) = 33
A31 = (– 1)4 (– 1) = – 1
A32 = (– 1)5 (– 3) = 3
A33 = (– 1)6.0 = 0
∴ Matrix of co-factors
= `[(-6, -15, 0),(0, 33, 33),(-1, 3, 0)]`
adj.A = `[(-6, 0, -1),(-15, 33, 3),(0, 33, 0)]`
A–1 = `1/|A|` (adj A)
∴ A–1 = `1/33 [(-6, 0, -1),(-15, 33, 3),(0, 33, 0)]`
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