Let A be a matrix $$A=\begin{bmatrix} *& *& 0& 0& *\\ *& *& *& 0& 0\\ 0& *& *& *& 0\\ 0& 0& *& *& *\\ *& 0& 0& *& *\ \end{bmatrix}$$
that has an $LU $decomposition, i.e. $A=LU$. Which entries in $L$ and $U$ may be zero and which may be non-zero$?
My attempt: I know that if $B$ is tridiagonal and has $LU$ decomposition, then the left lower triangular matrix $L$ will be $$L=\begin{bmatrix} 1& 0& 0& 0& 0\\ *& 1& 0& 0& 0\\ 0& *& 1& 0& 0\\ 0& 0& *& 1& 0\\ 0& 0& 0& *& 1 \end{bmatrix}$$ and right upper triangular matrix will be $$U=\begin{bmatrix} *& *& 0& 0& 0\\ 0& *& *& 0& 0\\ 0& 0& *& *& 0\\ 0& 0& 0& *& *\\ 0& 0& 0& 0& * \end{bmatrix}$$.
In general case $L$ is just a left lower triangular matrix with 1 on the main diagonal and $U$ is the right upper triangular matrix. However in this problem is there a special $LU$ decomposition for the matrix A like in the tridiagonal case?