At your school book fair, you buy 3 paperback and 2 hardcovers. Your best friend buys 2 paperbacks and 4 hardcovers. What is the inverse of the matrix represented by this situation?
Inverse of a Matrix
Recall that the multiplicative inverse of a real number is the reciprocal of the number and that the product of a number and its multiplicative inverse is the multiplicative identity, or 1. For example: . Now we need to define a multiplicative identity and a multiplicative inverse for a square matrix. For real numbers, 1 is considered the identity because we can multiply any number, , by 1 and the result is . In other words, the value of the number does not change. For matrices, the multiplicative inverse of a square matrix will be a square matrix in which the values of the main diagonal are 1 and the remaining values are all zero. The following are examples of identity matrices.
The products below illustrate how we can multiply a matrix by the identity and the result will be the original matrix.
Given:
In fact, it does not matter which order we multiply by the identity matrix. In other words, .
Now that we have defined an identity matrix, we can determine an inverse matrix such that .
The formula for finding the Inverse of a matrix is:
Given:
where
Note: If or , the matrix is called singular. The inverse of a singular matrix cannot be determined.
Let's find the inverse of the following matrices and verify our solution is correct (if the inverse exists).
First, use the formula above to find the inverse.
To verify that this is indeed the inverse, we must show that the product of the inverse and the original matrix is the identity matrix for a matrix. It will be easier to find this product using the form of the inverse where the reciprocal of the determinant has not been distributed inside the matrix as shown below:
Are and inverses?
If the matrices are inverses then their product will be the identity matrix.
Since the product is the identity matrix, the matrices are inverses of one another.
Use the formula above to find the inverse.
Examples
Example 1
Earlier, you were asked to find the inverse of the matrix that represents the book fair purchases.
The matrix that represents this situation is:
Use the formula you learned in this lesson to find the inverse.
Example 2
Are matrices and inverses of each other?
Yes, they are inverses.
Example 3
Find the inverse of
Example 4
Find the inverse of
Review
Determine whether the following pairs of matrices are inverses of one another.
 .

 and
 .

 and
 .

 and
 .

 and
Find the inverse of each matrix below, if it exists.
 .
 .
 .
 .
 .
 .
 .
 .
 .
 .
 For two 2x2 matrices, A and B, to be inverses of each other, what must be true of AB and BA?
Answers for Review Problems
To see the Review answers, open this PDF file and look for section 4.9.