Show that the LU decomposition of matrices of the form $\left[\begin{smallmatrix}0& x\\0 & y\end{smallmatrix}\right]$ is not unique How can I show that every matrix of the form $\begin{bmatrix}0& x\\0 & y\end{bmatrix}$ has an $LU$ factorization and that even if $L$ is unit lower triangular there is not a unique factorization?
I am completely stuck. Any ideas would be greatly appreciated

Suppose $L = \begin{bmatrix}1 & 0 \\ \ell & 1\end{bmatrix}$ and $U = \begin{bmatrix}u_1 & u_2 \\ 0 & u_3 \end{bmatrix}$. Then
$$
LU = \begin{bmatrix}u_1 & u_2\\u_1\ell & u_2\ell + u_3\end{bmatrix}.
$$
You want this to match with $\begin{bmatrix}0 & x\\0 & y\end{bmatrix}$. That means
\begin{align*}
u_1 & = 0 \\
u_1\ell & = 0 \\
u_2 & = x \\
u_2\ell + u_3 & = y.
\end{align*}
There are four unknowns and four equations, but the second equation is dependent on the first. That leaves one degree of freedom. $u_1$ and $u_2$ are fixed, but $u_3$ and $\ell$ can vary. $\ell$ can be chosen arbitrarily, and $u_3 = y - x\ell$ will work.
