$3\mathrm{.}10\mathrm{.}7.$ An$\backslash$times n is called a band matrix if aij $=0$ whenever $|$i $$ j$|>$ w for

some positive integer w$,$ called the bandwidth. In other words, the

nonzero entries of A are constrained to be in a band of w diagonal lines

above and below the main diagonal. For example, tridiagonal matrices

have bandwidth one, and diagonal matrices have bandwidth zero. If

A is a nonsingular matrix with bandwidth w$,$ and if A has an LU

factorization A $=$ LU$,$ then L inherits the lower band structure of A$,$

and U inherits the upper band structure in the sense that L has $$lower

bandwidth$$ w$,$ and U has $$upper bandwidth$$ w$.$ Illustrate why this

is true by using a generic $5\backslash$times $5$ matrix with a bandwidth of w $=2.$