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Question
Solve the following :
If A = `[(2, -3),(3, -2),(-1, 4)],"B" = [(-3, 4, 1),(2, -1, -3)]`, verify (A + 2BT)T = AT + 2B.
Solution
A = `[(2, -3),(3, -2),(-1, 4)] "and B" = [(-3, 4, 1),(2, -1, -3)]`
∴ AT = `[(2, 3, -1),(-3, -2, 4)] "and B"^"T" = [(-3, 2),(4, -1),(1, -3)]`
∴ A + 2BT = `[(2, -3),(3, -2),(-1, 4)] + 2[(-3, 2),(4, -1),(1, -3)]`
= `[(2, -3),(3, -2),(-1, 4)] + [(-6, 4),(8, -2),(2, -6)]`
= `[(2 - 6, -3 + 4),(3 + 8 , -2 - 2),(-1 - 2, 4 - 6)]`
∴ A + 2BT = `[(-4, 1),(11, -4),(1, -2)]`
∴ *A + 2BT)T = `[(-4, 11, 1),(1, -4, -2)]` ...(i)
A + 2B = `[(2, 3, -1),(-3, -2, 4)] + 2[(-3, 4, 1),(2, -1, -3)]`
= `[(2, 3, -1),(-3, -2, 4)] + [(-6, 8, 2),(4, -2, -6)]`
= `[(-4, 11, 1),(1, -4, -2)]` ...(iii)
From (i) and (ii), we get
(A + 2BT)T = AT + 2B.
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