Question

if A = `[(3, -2),(4,-2)] and l = Matric [(1,0),(0,1)]`  find k so that `A^2 = kA - 2I`

Answer 1

Given: A = `[(3, -2), (4,-2)],` I = `[(1,0),(0,1)]`

`"A"^2 = "A". "A" = [(3, -2),(4,-2)] [(3, -2),(4,-2)]`

`= [(9 - 8, -6 + 4),(12 - 8, -8 + 4)]`

`= [(1,-2),(4,-4)]`

KA - 2I = K `[(3, -2),(4,-2)] - 2 [(1,0),(0,1)]`

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

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

`"A"^2 = "KA" - 2"I"`

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

3k - 2 = 1

⇒ 3k = 3

k = 1