Determinants

If A be any given square matrix, then \(A(adj A) = (adj A)A = |A|I\), where I is the identity matrix of the same order A.

Verification Let \(A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}\), then \(adj A = \begin{bmatrix} A_{11} & A_{21} & A_{31} \\ A_{12} & A_{22} & A_{32} \\ A_{13} & A_{23} & A_{33} \end{bmatrix}\) Since sum of product of elements of a row with their corresponding cofactors is equal to \(|A|\) and otherwise zero, we have \(A(adj A) = \begin{bmatrix} |A| & 0 & 0 \\ 0 & |A| & 0 \\ 0 & 0 & |A| \end{bmatrix} = |A|\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = |A|I\) Similarly, we can verify \((adj A) A = |A| I\) Hence A (adj A) = (adj A) A = |A| I

Definition 4 A square matrix A is said to be singular if \(|A| = 0\). Definition 5 A square matrix A is said to be non-singular if \(|A| \neq 0\).

We state the following theorems without proof. Theorem 2 If A and B are nonsingular matrices of the same order, then AB and BA are also nonsingular matrices of the same order. Theorem 3 The determinant of the product of matrices is equal to product of their respective determinants, that is, \(|AB| = |A| |B|\), where A and B are square matrices of the same order.