\subsection{Problem 11}

Compute the LU factorization for

\begin{equation*}
    \matr{A} = 
    \begin{bmatrix}
    \phantom{-}1 & \phantom{-}3 & 3 & 2 \\
    \phantom{-}2 & \phantom{-}6 & 9 & 5 \\
    -1 & -3 & 3 & 0
    \end{bmatrix}
\end{equation*}




Determine a set of basic variables and a set of free variables, and find a homogeneous solution to $\mathbf{Ax = 0}$. What is the rank of $\mathbf{A}$?