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4.7 Inverse Matrix

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4.7 Inverse Matrix
1. If A is a square matrix, B is another square matrix and A × B = B × A = I, then matrix A is the inverse matrix of matrix B and vice versa. Matrix A is called the inverse matrix of B for multiplication and vice versa.
 
2. The symbol A-1 denotes the inverse matrix of A.

3. Inverse matrices can only exist for square matrices but not all square matrices have inverse matrices.

4. If AB ≠ I or BA ≠ I, then A is not the inverse of B and B is not the inverse of A.

Example 1:
Determine whether matrix  A = ( 2 9 1 5 )  is an inverse matrix of matrix B = ( 5 − 9 − 1 2 ) .

Solution:
AB=( 2 9 1 5 )( 5 −9 −1 2 ) =( 2×5+9×−1 2×−9+9×2 1×5+5×−1 1×−9+5×2 ) =( 10+( −9 ) −18+18 5+( −5 ) −9+10 ) =( 1 0 0 1 )=I AB=( 5 −9 −1 2 )( 2 9 1 5 ) =( 5×2+( −9 )×1 5×9+( −9 )×5 −1×2+2×1 −1×9+2×5 ) =( 10+( −9 ) 18−18 −2+2 −9+10 ) =( 1 0 0 1 )=I AB=BA=I ∴A is the inverse matrix of B and vice versa.


5. The inverse of a matrix may also be found using a formula.
If A = ( a b c d ) , then the inverse matrix of A, A-1, is given by the formula below.
     A −1 = 1 ad−bc ( d −b −c a ), where ad−bc≠0    
6. ad – bc is known as the determinant of matrix A.

7. If the determinant, ad – bc = 0, then the inverse matrix of A does not exist.

Example 2:
Find the inverse matrix of A = ( 6 1 − 9 − 1 )  using the formula.

Solution:
A = ( 6 1 − 9 − 1 ) a = 6 , b = 1 , c = − 9 , d = − 1 A − 1 = 1 a d − b c ( d − b − c a ) A − 1 = 1 6 × − 1 − ( 1 × − 9 ) ( − 1 − 1 9 6 ) A − 1 = 1 − 6 + 9 ( − 1 − 1 9 6 ) A − 1 = 1 3 ( − 1 − 1 9 6 ) = ( − 1 3 − 1 3 3 2 )


Example 3:
The inverse matrix of ( − 7 2 − 9 2 ) is r ( 2 s 9 t ) .  Find the value of r, of s and of t.

Solution:
Let A = ( − 7 2 − 9 2 ) A − 1 = 1 − 7 × 2 − ( − 9 ) × 2 ( 2 − 2 9 − 7 ) A − 1 = 1 4 ( 2 − 2 9 − 7 ) ∴ r ( 2 s 9 t ) = 1 4 ( 2 − 2 9 − 7 ) By comparison, r = 1 4 , s = − 2 , t = − 7.