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Question
If A = `[(1, 4), (1, -3)]` and B = `[(1, 2),(-1, -1)]`, find:
- (A + B)2
- A2 + B2
- Is (A + B)2 = A2 + B2 ?
Solution
i.
A + B = `[(1, 4),(1, -3)] + [(1, 2),(-1, -1)]`
= `[((1 + 1, 4 + 2)),((1 - 1, - 3 - 1))]`
= `[(2, 6),(0, -4)]`
Now, (A + B)2 = (A + B)(A + B)
= `[(2, 6),(0, -4)] [(2, 6),(0, -4)]`
= `[((2)(2) + (6)(0), (2)(6) + (6)(-4)), ((0)(2) + (-4)(0), (0)(6) + (-4)(-4))]`
= `[(4, -12),(0, 16)]`
ii.
A2 = `[(1, 4), (1, -3)][(1, 4), (1, -3)]`
= `[(1 + 4, 4 - 12),(1 - 3, 4 + 9)]`
= `[(5, -8),(-2, 13)]`
B2 = `[(1, 2),(-1, -1)][(1,2),(-1, -1)]`
`[(1 -2, 2 - 2),(-1 + 1, -2 + 1)]`
`[(-1, 0),(0, -1)]`
A2 + B2 = `[(5, -8),(-2, 13)] + [(-1, 0),(0, -1)]`
A2 + B2 = `[(4, -8),(-2, 12)]`
iii. No, (A + B)2 ≠ A2 + B2.
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