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Question
If A = `[(2, -1),(3, -2),(4, 1)] "and B" = [(0, 3, -4),(2, -1, 1)]`, verify that (AB)T = BTAT.
Solution
A = `[(2, -1),(3, -2),(4, 1)] "and B" = [(0, 3, -4),(2, -1, 1)]`
∴ AT = `[(2, 3, 4),(-1, -2, 1)] "and B"^"T" = [(0, 2),(3, -1),(-4, 1)]`
AB = `[(2, -1),(3, -2),(4, 1)][(0, 3, -4),(2, -1, 1)]`
= `[(0 - 2, 6 + 1, -8 - 1),(0 - 4, 9 + 2, -12 - 2),(0 + 2, 12 - 1, -16 + 1)]`
= `[(-2, 7, -9),(-4, 11, -14),(2, 11, -15)]`
∴ (AB)T = `[(-2, -4, 2),(7, 11, 11),(-9, -14, -15)]` ...(i)
BTAT = `[(0, 2),(3, -1),(-4, 1)][(2, 3, 4),(-1, -2, 1)]`
= `[(0 - 2, 0 - 4, 0 + 2),(6 + 1, 9 + 2, 12 - 1),(-8 - 1, -12 - 2, -16 + 1)]`
= `[(-2, -4, 2),(7, 11, 11),(-9, -14, -15)]` ...(ii)
From (i) and (ii), we get
(AB)T = BTAT.
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