Since sum of product of elements of a row (or a column) with corresponding cofactors is equal to |A| and otherwise zero, we have DETERMINANTS A ( adj A) = = A = A I Similarly, we can show ( adj A) A = A I Hence A ( adj A) = ( adj A) A = A I Definition A square matrix A is said to be singular if A = . For example, the determinant of matrix A = is zero Hence A is a singular matrix. Definition A square matrix A is said to be non-singular if A ≠ Let A = . Then A = = – = – ≠ .
Hence A is a nonsingular matrix We state the following theorems without proof. Theorem If A and B are nonsingular matrices of the same order, then AB and BA are also nonsingular matrices of the same order. Theorem 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 Remark We know that ( adj A) A = A I = ≠ , Writing determinants of matrices on both sides, we have ( A)A adj i.e. |( adj A)| |A| = (Why?) i.e.
|( adj A)| |A| = |A| ( ) i.e. |( adj A)| = | A | In general, if A is a square matrix of order n , then | adj (A)| = |A| n – . Theorem A square matrix A is invertible if and only if A is nonsingular matrix. Proof Let A be 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 = (since I , AB A B ) This gives A ≠ . Hence