1.) If $\displaystyle X = P^{-1}AP$ and $\displaystyle A^3 = I$, prove that $\displaystyle X^3 = I$

Since order of multiplication don't matter when you're multiplying three matrices and more ( I deduced this from the multiplication of matrices (AB)C=A(BC) ),

X = IA

X = A

Multiplying A^2 on both sides,

A^2X = A^3

A^2X = I

This is where I'm stuck..

2.) For A = $\displaystyle \begin{bmatrix} 2 & 1 & -1 \\-1 & 2 & 1 \\0 & 6 & 1 \end{bmatrix}$ and B = $\displaystyle \begin{bmatrix} 4 & 7 & -3 \\-1 & -2 & 1 \\6 & 12 & -5 \end{bmatrix}$

calculate AB and hence solve the system of equations

4a+7b-3c = -8

-a-2b+c = 3

6a+12b-5c = -15

I got the identity matrix for AB.But I do not know how to use it to solve the system of equations, I had applied the identity matric to the system of equations so I got 4a=-8, -2b = 3 and -5c = -15, but this does not seem to be the way..

Thank you for your time!