# 4.7 Inverse Matrix

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 $A=\left(\begin{array}{cc}2& 9\\ 1& 5\end{array}\right)$  is an inverse matrix of matrix $B=\left(\begin{array}{cc}5& -9\\ -1& 2\end{array}\right).$

Solution:

5. The inverse of a matrix may also be found using a formula.
If $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ , then the inverse matrix of A, A-1, is given by the formula below.

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 $A=\left(\begin{array}{cc}6& 1\\ -9& -1\end{array}\right)$  using the formula.

Solution:

Example 3:
The inverse matrix of  Find the value of r, of s and of t.

Solution: