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Question
If A = `[(1, 2, 0),(-2, -1, -2),(0, -1, 1)]`, find A–1. Using A–1, solve the system of linear equations x – 2y = 10, 2x – y – z = 8, –2y + z = 7.
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Solution
We have, A = `[(1, 2, 0),(-2, -1, -2),(0, -1, 1)]`
Co-factors are:
A11 = –3
A12 = 2
A13 = 2
A31 = –4
A32 = 2
A33 = 3
∴ adjA = `[(-3, 2, 2),(-2, 1, 1),(-4, 2, 3)]^"T"`
= `[(-3, -2, -4),(2, 1, 2),(2, 1, 3)]`
|A| = 1(–3) – 2(–2) + 0 = 1
∴ `"A"^-1 ("adj A")/|"A"| = [(-3, -2, -4),(2, 1, 2),(2, 1, 3)]`
Now the system of linear equations is
x – 2 = 10
2x– y – z = 8
And –2y + z = 7
or AX = B
i.e., `[(1, -2, 0),(2, -1, -1),(0, -2, 1)][(x),(y),(z)] = [(10),(8),(7)]`
Where, A = `[(1, -2, 0),(2, -1, -1),(0, -2, 1)]`
X = `[(x),(y),(z)]` and B + `[(10),(8),(7)]`
∴ X = `"A"^-1"B"`
⇒ `[(x),(y),(z)] = [(-3, 2, 2),(-2, 1, 1),(-4, 2, 3)] [(10),(8),(7)]`
= `[(-30 + 16 + 14),(-20 +8 + 7),(-40 + 16 + 21)]`
= `[(0),(-5),(-3)]`
∴ x = 0, y = –5 and = –3
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