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Question
If A = `[(1, 0),(-1, 7)]`, find k so that A2 – 8A – kI = O, where I is a unit matrix and O is a null matrix of order 2.
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Solution
A2 = A · A = `[(1, 0),(-1, 7)] [(1, 0),(-1, 7)]`
= `[(1 - 0, 0 + 0),(-1 - 7, 0 + 49)]`
= `[(1, 0),(-8, 49)]`
∴ A2 – 8A – kI = `[(1, 0),(-8, 49)] - 8 [(1, 0),(-1, 7)] -"k"[(1, 0),(0, 1)]`
= `[(1, 0),(-8, 49)] - [(8, 0),(-8, 56)] - [("k", 0),(0, "k")]`
= `[(1 - 8 - "k", 0 - 0 - 0),(-8 + 8 - 0, 49 - 56 - "k")]`
= `[(-7 - "k", 0),(0, -7 - "k")]`
But A2 – 8A – kI = 0
∴ `[(-7 - "k", 0),(0, -7 - "k")] = [(0, 0),(0, 0)]`
∴ –7 – k = 0
∴ k = – 7.
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