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.

Answer 1

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