## Find a Nonsingular Matrix Satisfying Some Relation

## Problem 280

Determine whether there exists a nonsingular matrix $A$ if

\[A^2=AB+2A,\]
where $B$ is the following matrix.

If such a nonsingular matrix $A$ exists, find the inverse matrix $A^{-1}$.

**(a)** \[B=\begin{bmatrix}

-1 & 1 & -1 \\

0 &-1 &0 \\

1 & 2 & -2

\end{bmatrix}\]

**(b)** \[B=\begin{bmatrix}

-1 & 1 & -1 \\

0 &-1 &0 \\

2 & 1 & -4

\end{bmatrix}.\]