Let $\mathbf{A}$ be a matrix, and let $\mathbf{x}$ and $\mathbf{y}$ be linearly independent vectors such that $\mathbf{A} \mathbf{x} = \mathbf{y}, \mathbf{A} \mathbf{y} = \mathbf{x} + 2\mathbf{y}.$Then we have that $\mathbf{A}^{-1} \mathbf{x} = a \mathbf{x} + b\mathbf{y}$for some scalars $a$ and $b$.
So, currently I have been going in a loop by plugging in the matrix a into a vector (x;y). The quantity gotten would be plugged into Ay = x+2y. I have no clue on how to solve it. So confused.
Apply $A^{-1}$ to the left side of both equations to get $$ x = A^{-1} y, \quad y = A^{-1} x + 2 A^{-1} y $$ Can you rearrange from here?