Theorem: Let $B$ be an $n \times n$ matrix and let $E$ be an $n \times n$ elementary matrix.
Then
\(det(EB) = det(E)det(B)\)
Proof: trivial
Theorem: A square matrix is invertible iff $det(A) \neq 0$
Proof:
Let A be an $n \times n$ matrix and $R$ be the reduced row echelon form of $A$.
Then we know that we can represent it as
\(E_r ... E_2 E_1 A = R\)
where $E_i$ are the elementary matrices corresponding to the elemantry transformations needed to convert $A$ to $R$
Taking the determinant on both sides, and by repeated application of the previous theorem, we have \(det(E_r) ... det(E_2) det(E_1) det(A) = det(R)\) Thus we get