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Question
If A = `[(2, -3),(5, -4),(-6, 1)], "B" = [(2, 1),(4, -1),(-3, 3)], "C" = [(1, 2),(-1, 4),(-2, 3)]`, then show that (A + B)T = AT + BT.
Solution
A = `[(2, -3),(5, -4),(-6, 1)] + [(2, 1),(4, -1),(-3, 3)]`
= `[(2 + 2, -3 + 1),(5 + 4, -4 - 1),(-6 - 3, 1 + 3)]`
= `[(4, -2),(9, -5),(-9, 4)]`
∴ (A + B)T = `[(4, 9, -9),(-2, -5, 4)]` ...(i)
Now, AT = `[(2, 5, -6),(-3, -4, 1)] "and" "B"^"T" = [(2, 4, -3),(1, -1, 3)]`
∴ AT + BT = `[(2, 5, -6),(-3, -4, 1)] + [(2, 4, -3),(1, -1, 3)]`
= `[(2 + 2, 5 + 4, -6 - 3),(-3 + 1, -4 - 1, 1 + 3)]`
= `[(4, 9, -9),(-2, -5, 4)]` ...(ii)
From (i) and (ii, we get
(A + B)T = AT + BT.
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