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