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Question
For the matrices A and B, verify that (AB)′ = B'A' where A = `[(0),(1),(2)]`, B = `[(1, 5, 7)]`
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Solution
Given, A = `[(0),(1),(2)]` and B = `[(1, 5, 7)]`
So, AB = `[(0),(1),(2)] xx [(1, 5, 7)]`
= `[(0 xx 1, 0 xx 5, 0 xx 7),(1 xx 1, 1 xx 5, 1 xx 7),(2 xx 1, 2 xx 5, 2 xx 7)]`
= `[(0, 0, 0), (1, 5, 7),(2, 10, 14)]`
Now, (AB)' = `[(0, 1, 2),(0, 5, 10),(0, 7, 14)]` ...(i)
So, A' = `[(0, 1, 2)]` and B' = `[(1),(5),(7)]`
Now, B'A' = `[(1),(5),(7)] xx [(0, 1, 2)]`
= `[(1 xx 0, 1 xx 1, 1 xx 2), (5 xx 0, 5 xx 1, 5 xx 2), (7 xx 0, 7 xx 1, 7 xx 2)]`
= `[(0, 1, 2),(0, 5, 10),(0, 7, 14)]` ...(ii)
Equations (i) and (ii) prove that,
∴ (AB)' = B'A'
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