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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)]`
= `[(-5, 3, -2),(6, 9, 9),(-1, 4, 2)]`
Now, (A + B)' = `[(-5, 6, -1),(3, 9, 4),(-2, 9, 2)]` ...(i)
A' = `[(-1, 5, -2),(2, 7, 1),(3, 9, 1)]` and B' = `[(-4, 1, 1),(1, 2, 3),(-5, 0, 1)]`
Then, 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)]`
`[(-5, 6, -1),(3, 9, 4),(-2, 9, 2)]` ...(ii)
Equations (i) and (ii) prove that,
(A + B)' = A' + B'
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