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Question
If A = `[(-3, -2, -4),(2, 1, 2),(2, 1, 3)]`, B = `[(1, 2, 0),(-2, -1, -2),(0, -1, 1)]` then find AB and use it to solve the following system of equations:
x – 2y = 3
2x – y – z = 2
–2y + z = 3
Solution
Given, A = `[(-3, -2, -4),(2, 1, 2),(2, 1, 3)]`
and B = `[(1, 2, 0),(-2, -1, -2),(0, -1, 1)]`
Now, AB = `[(-3, -2, -4),(2, 1, 2),(2, 1, 3)][(1, 2, 0),(-2, -1, -2),(0, -1, 1)]`
= `[(-3 + 4 + 0, -6 + 2 + 4, 0 + 4 - 4),(2 - 2 + 0, 4 - 1 - 2, 0 - 2 + 2),(2 - 2 + 0, 4 - 1 - 3, 0 - 2 + 3)]`
= `[(1, 0, 0),(0, 1, 0),(0, 0, 1)]`
∴ AB = I
(AB)B–1 = IB–1
A = B–1 ...(i)
Now, equations are
x – 2y = 3
2x – y – z = 2
– 2y + z = 3
∴ `[(1, -2, 0),(2, -1, -1),(0, -2, 1)][(x),(y),(z)] = [(3),(2),(3)]`
CX = D
Here, C = BT ...(ii)
∴ C–1(CX) = C–1 D
`\implies` IX = C–1 D
`\implies` X = C–1 D
= [BT]–1 D ...[From (ii)]
= [B–1]T D
= [A]T D ...[From (i)]
X = `[(-3, -2, -4),(2, 1, 2),(2, 1, 3)]^T[(3),(2),(3)]`
X = `[(-3, 2, 2),(-2, 1, 1),(-4, 2, 3)][(3),(2),(3)]`
= `[(-9 + 4 + 6),(-6 + 2 + 3),(-12 + 4 + 9)]`
`[(x),(y),(z)] = [(1),(-1),(1)]`
Hence x = 1, y = – 1 and z = 1.
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