Monomial matrices (Generalized Permutation matrices )

Question

We define Monomial matrices as following

A matrix is said to be monomial if each row and column has exactly one non-zero entry. Let $N$ denote the set of monomial matrices

I want to show that it is a subgroup of $GL_{n}(F)$.

I have problem in showing that if $A\in N$ then its inverse also belongs to $N$.

Answer 1

Hint: The following are equivalent:

• $A \in N$
• $A = DP$ for some (invertible) diagonal matrix $D$ and permutation matrix $P$
• $A = PD$ for some (invertible) diagonal matrix $D$ and permutation matrix $P$