1. (30 pts) Given matrix $A = \begin{bmatrix} 2 & -1 \ 3 & 2 \end{bmatrix}$ has been decomposed into an upper triangular matrix $U = \begin{bmatrix} 3 & 2 \ 0 & -7/3 \end{bmatrix}$ and a lower triangular matrix $L = \begin{bmatrix} 1 & 0 \ 2/3 & 1 \end{bmatrix}$ with the permutation matrix $P = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}$ such that $LU = PA$, find the matrix inverse $A^{-1}$ using L, U, and P and then use $A^{-1}$ to solve for the unknown vectors $\begin{bmatrix} x_1 \ x_2 \end{bmatrix}$ of the system of equations below. Please show your work.
$\begin{cases} 2x_1 - x_2 = 3 \ 3x_1 + 2x_2 = 8 \end{cases}$
