Note that the determinant of $A$ is $-2$, so your matrix is invertible i.e. there exists some matrix $B$ such that $AB=BA=I$.

Note that the determinant of $A$ is $-2$, so your matrix is invertible i.e. there exists some matrix $B=A^{-1}$ such that $AB=BA=I$. If I calculated this correctly $$B = \begin{pmatrix} 1 & 1/2& -1/2\\ 1 & -2 &1 \\ -1 & 3/2 & -1/2 \end{pmatrix}.$$