A square matrix \(\Mtrx{A}\) is invertible exactly when \(\DtrmnntOf{ \Mtrx{A} } \neq 0\).
A square matrix \(\Mtrx{A}\) is invertible exactly when \(\DtrmnntOf{ \Mtrx{A} } \neq 0\).
We know that \(\DtrmnntOf{ \Mtrx{A} }\neq 0\) if and only if any upper triangular reduction matrix \(\Mtrx{U}\) of \(\Mtrx{A}\) has only nonzero elements \(d_1,\dots ,d_n\) on the diagonal. This happens if and only if the RREF of \(\Mtrx{U}\) and, hence, of \(\Mtrx{A}\) is the identity matrix. This in turn is equivalent to the invertibility of \(\Mtrx{A}\).