Question

Find k, if A = `[(3, -2),(4, -2)]` and A² = kA – 2I.

Answer 1

A² = kA – 2I
∴ A.A + 2I = kA

∴ `[(3, -2),(4, -2)][(3, -2),(4, -2)] + 2[(1, 0),(0, 1)] = "k"[(3, -2),(4, -2)]`

∴ `[(9 - 8 ,-6 + 4),(12 - 8, -8 + 4)] + [(2, 0),(0, 2)] = [(3"k", -2"k"),(4"k", -2"k")]`

∴ `[(1, -2),(4, -4)] + [(2, 0),(0, 2)] = [(3"k", -2"k"),(4"k", -2"k")]`

∴ `[(1 + 2, -2 + 0),(4 + 0, -4 + 2)] = [(3"k", -2"k"),(4"k", -2"k")]`

∴ `[(3, -2),(4, -2)] = [(3"k", -2"k"),(4"k", -2"k")]`

∴ By equality of matrices, we get
3k = 3
∴ k = 1.