Express A^(-1) matrix as linear combination of A^2, A and I3

Question

I am a bit confused. How can I express A^(-1) matrix as linear combination of A^2, A and I3?

This is the matrix A: [0, 0, 7], [1, 0, 3], [0, 1, 8]

I found A^2 to be [0, 7, 56], [0, 3, 31], [1, 8, 67]

Thanks

Answer 1

$$ A-\lambda I=\left[\begin{array}{ccc}-\lambda & 0 & 7 \\ 1 & \lambda & 3 \\ 0 & 1 & 8-\lambda \end{array}\right] $$ The characteristic polynomial is \begin{align} \mbox{det}(A-\lambda I)&=-\lambda^2(8-\lambda)+7-(-3\lambda)\\ &=\lambda^3-8\lambda^2+3\lambda+7 \end{align} Therefore $$ A^3-8A^2+3A=-7I \\ A(A^2-8A+3I)=-7I \\ A^{-1}=-\frac{1}{7}(A^2-8A+3I). $$