7. (12 points) Let

$A = \begin{bmatrix} 0 & -3 & 3 & 9 \\ 2 & 3 & -1 & 1 \\ 1 & 3 & -2 & 2 \\ 4 & 10 & -5 & 1 \end{bmatrix}$

a. (6 points) Find the $A^{-1} = ?$

b. (6 points) Find the $PA=LU$ factorization for $A$, where $P$ is a permutation matrix, $L$ is a lower triangular matrix with unit diagonal, and $U$ is an upper triangular matrix.