Question

If A = `[(1 , 0),(-1 ,7)]` and I = `[(1 , 0),(0 ,1)]`, then find k so that A² = 8A + kI.

Answer 1

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.