Question

If A= `((1,0,2),(0,2,1),(2,0,3))` and A³ - 6A² +7A + kI₃ = O find k.

Answer 1

`A=[(1,0,2),(0,2,1),(2,0,3)]`

`A^2="AA" = [(1,0,2),(0,2,1),(2,0,3)][(1,0,2),(0,2,1),(2,0,3)]`

`=[(5,0,8),(2,4,5),(8,0,13)]`

`A^3=A^2.A=[(5,0,8),(2,4,5),(8,0,13)][(1,0,2),(0,2,1),(2,0,3)]`

`=[(21,0,34),(12,8,23),(34,0,55)]`

∴ A³ - 6A² + 7A + KI₃ = 0

`=>[(21,0,34),(12,8,23),(34,0,55)]-6[(5,0,8),(2,4,5),(8,0,13)]+7[(1,0,2),(0,2,1),(2,0,3)]+k[(1,0,0),(0,1,0),(0,0,1)]=0`

⇒ k = 2