. . Definition of inverse matrix of a square matrix Now, we define the inverse of a square matrix. Definition .
Let A be a square matrix of order n .If there exists a square matrix B of order n such that AB BA I n , then the matrix B is called an inverse of A . Theorem . If a square matrix has an inverse, then it is unique. Proof Let A be a square matrix order n such that an inverse of A exists.
If possible, let there be two inverses B and C of A .Then, by definition, we have AB BA I n and AC CA I n Using these equations, we get C CI C AB CA B I B B Hence the uniqueness follows. Notation The inverse of a matrix A is denoted by A − . Note AA A A I n - - Applications of Matrices and Determinants Theorem . Let A be square matrix of order n .Then, A − exists if and only if A is non-singular.
Proof Suppose that A − exists. Then AA A A I n By the product rule for determinants, we get det( det( )det( det( )det( ) det( AA I n So, A ≠ det( ) . Hence A is non-singular. Conversely, suppose that A is non-singular.
Then A ¹ . By Theorem . , we get A A A I n (adj (adj So, dividing by A , we get A A A I n adj adj = Thus, we are able to find a matrix B = adj such that AB BA I n Hence, the inverse of A exists and it is given by A −− == adj . Remark The determinant of a singular matrix is and so a singular matrix has no inverse.
Example . If A = is non-singular, find A − . We first find adj A . By definition, we get adj A M M M M T T = + = = Since A is non-singular, A ad bc ≠ .
As A − = adj , we get A ad bc − = Example . Find the inverse of the matrix Let A = . Then | ( ) A = + − = −≠ Therefore, A − exists. Now, we get - - adj A = − − + − − − + − + − −− T T Hence, A − = | ( adj