(A) Let $A$ and $B$ be $n \times n$ matrices with entries in $\mathbb{N}$. Show that if $B = A^{-1}$ then $A$ and $B$ are permutation matrices. (A permutation matrix is a matrix obtained by permuting the rows of the identity matrix.)\\
(B) Let $A$ be an $n \times n$ complex matrix that is not a scalar multiple of $I_n$. Show that $A$ is similar to a matrix $B$ such that $B_{1,1}$ (i.e. the top left entry of $B$) is 0.