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Question
If A = `[(1 , 0),(-1 ,7)]` and I = `[(1 , 0),(0 ,1)]`, then find k so that A2 = 8A + kI.
Solution
We have
A = `[(1 , 0),(-1 ,7)]`
A2 = AA = `[(1 , 0),(-1 ,7)][(1 , 0),(-1 ,7)]`
= `[(1 , 0),(-8 ,49)]`
And 8A + kI = 8 `[(1 , 0),(-1 ,7)] +k [(1 , 0),(0 ,1)]`
= `[(8 , 0),(-8 , 56)] + [(k , 0),(0 , k)]`
= `[(8 + k , 0),(-8 , 56 + k)]`
Thus A2 = 8A + kI
⇒ `[(1 , 0),(-8 ,49)] = [(8 + k , 0),(-8 , 56 + k)]`
⇒ 1 = 8 + k
⇒ k = -7
Also 56 + k = 49
⇒ k = -7.
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