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Question
Solve the following :
If A = `[(2, -4),(3, -2),(0, 1)], "B" = [(1, -1, 2),(-2, 1, 0)]`, then show that (AB)T = BTAT.
Solution
AB = `[(2, -4),(3, -2),(0, 1)] [(1, -1, 2),(-2, 1, 0)]`
= `[(2 + 8, -2 - 4, 4 + 0),(3 + 4, -3 - 2, 6 - 0),(0 - 2, 0 + 1, 0 + 0)]`
= `[(10, -6, 4),(7, -5, 6),(-2, 1, 0)]`
∴ (AB)T = `[(10, 7, -2),(-6, -5, 1),(4, 6, 0)]` ...(i)
Now,AT = `[(2, 3, 0),(-4, -2, 1)] "and B"^"T" = [(1, -2),(-1, 1),(2, 0)]`
∴ BTAT = `[(1, -2),(-1, 1),(2, 0)] [(2, 3, 0),(-4, -2, 1)]`
= `[(2 + 8, 3 + 4, 0 - 2),(-2 - 4, -3 - 2, 0 + 1),(4 - 0, 6 - 0, 0 + 0)]`
∴ BTAT = `[(10, 7, -2),(-6, -5, 1),(4, 6, 0)]` ...(ii)
From (i) and (ii), we get
(AB)T = BTAT.
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