Determinants

A square matrix A is invertible if and only if A is nonsingular matrix. Proof Let A be an invertible matrix of order n and I be the identity matrix of order n. Then there exists a square matrix B of order n such that \(AB = BA = I\). Now \(AB = I\). So \(|AB| = |I|\) or \(|A| |B| = 1\) This gives \(|A| \neq 0\). Hence A is nonsingular. Conversely, let A be nonsingular. Then \(|A| \neq 0\). Now \(A(adj A) = (adj A)A = |A| I\) (Theorem 1) or \(A(\frac{1}{|A|}adj A) = (\frac{1}{|A|}adj A)A = I\) or \(AB = BA = I\), where \(B = \frac{1}{|A|}adj A\) Thus A is invertible and \(A^{-1} = \frac{1}{|A|}adj A\)