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Question
If A = `[(-1, 2, 3),(5, 7, 9),(-2, 1, 1)]` and B = `[(-4, 1, -5),(1, 2, 0),(1, 3, 1)]`, then verify that (A – B)' = A' – B'
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Solution
Given, A = `[(-1, 2, 3),(5, 7, 9),(-2, 1, 1)]` and B = `[(-4, 1, -5),(1, 2, 0),(1, 3, 1)]`
Then, (A – B) = A = `[(-1, 2, 3),(5, 7, 9),(-2, 1, 1)] - [(-4, 1, -5),(1, 2, 0),(1, 3, 1)]`
= `[(-1 + 4, 2 - 1, 3 + 5),(5 - 1, 7 - 2, 9 - 0), (-2 - 1, 1 - 3, 1 - 1)]`
= `[(3, 1, 8),(4, 5, 9),(-3, -2, 0)]`
Then, (A – B)' = `[(3, 1, 8),(4, 5, 9),(-3, -2, 0)] = [(3, 4, -3),(1, 5, -2),(8, 9, 0)] ` ...(i)
We know that, A' = `[(-1, 5, -2), (2, 7, 1),(3, 9, 1)]` and B' = `[(-4, 1, 1),(1, 2, 3),(-5, 0, 1)]`
A' – B' = `[(-1, 5, -2), (2, 7, 1),(3, 9, 1)] - [(-4, 1, 1),(1, 2, 3),(-5, 0, 1)]`
= `[(-1 + 4, 5 - 1, -2 - 1),(2 - 1, 7 - 2, 1 - 3),(3 + 5, 9 - 0, 1 - 1)]`
= `[(3, 4, -3),(1, 5, -2),(8, 9, 0)]` ...(ii)
Equations (i) and (ii) prove that,
(A – B)' = A' – B'
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