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Question
Solve the following system of linear equations by matrix inversion method:
2x + 3y – z = 9, x + y + z = 9, 3x – y – z = – 1
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Solution
`[(2, 3, -1),(1, 1, 1),(3, -1, -1)][(x),(y),(z)] = [(9),(9),(-1)]`
AX = B
X = `"A"^-1"B"`
A = `[(2, 3, -1),(1, 1, 1),(3, -1, -1)]`
|A| = 2(–1+1) –3(–1 – 3) –1(–1 – 3)
= 0 + 12 + 4
=16 ≠ 0 A-1 exists.
adj A = `[((-1 + 1), -(-1 - 3), (-1 - 3)),(-(-3 - 1),(-2 + 3), -(- 2 - 9)),((3 + 1), -(2 + 1), (2 - 3))]^"T"`
= `[(0, 4, -4),(4, 1, 11),(4, -3, -1)]^"T"`
= `[(0, 4, 4),(4, 1, -3),(-4, 11, -1)]`
`"A"^-1 = 1/|"A"|`
adj A = `1/16 [(0, 4, 4),(4, 1, -3),(-4, 11, -1)]`
X = `"A"^-1"B"`
`[(x),(y),(z)] = 1/16[(0, 4, 4),(4, 1, -3),(-4, 11, 1)][(9),(9),(-1)]`
= `1/16 [(0 + 36 - 4),(36 + 9 + 3),(- 36 + 99 + 1)]`
`[(x),(y),(z)] = 1/16 [(32), (48), (64)]`
= `[(2),(3),(4)]`
∴ x = 2, y = 3, z = 4
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