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Question
Answer the following question:
If A = `[(1, 2, 3),(2, 4, 6),(1, 2, 3)]`, B = `[(1, -1, 1),(-3, 2, -1),(-2, 1, 0)]`, show that AB and BA are both singular matrices
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Solution
AB = `[(1, 2, 3),(2, 4, 6),(1, 2, 3)] [(1, -1, 1),(-3, 2, -1),(-2, 1, 0)]`
= `[(1 - 6 - 6, -1 + 4 + 3, 1 - 2+ 0),(2 - 12 - 12, -2 + 8 + 6, 2 - 4 + 0),(1 - 6 - 6, -1 + 4 + 3, 1 - 2 + 0)]`
= `[(-11, 6, -1),(-22, 12, -2),(-11, 6, -1)]`
∴ |AB| = `|(-11, 6, -1),(-22, 12, -2),(-11, 6, -1)|`
= 0 ...[∵ R1 and R3 are identical]
∴ AB is a singular matrix
BA = `[(1, -1, 1),(-3, 2, -1),(-2, 1, 0)] [(1, 2, 3),(2, 4, 6),(1, 2, 3)]`
= `[(1 - 2 + 1, 2 - 4 + 2, 3 - 6 + 3),(-3 + 4 - 1, -6 + 8 - 2, -9 + 12 - 3),(-2 + 2 + 0, -4 + 4 + 0, -6 + 6 + 0)]`
= `[(0, 0, 0),(0, 0, 0),(0, 0, 0)]`
∴ |BA| = 0
∴ BA is a singular matrix.
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