# Lesson Explainer: Properties of Inverse Matrices Mathematics

In this explainer, we will learn how to use some properties of matrix inverse.

An matrix is said to be invertible if there exists an matrix such that the product of and is , where is the identity matrix:

If exists, we say that it is the inverse of , denoted .

Note that, as implied in this definition, a matrix must be square to be invertible, but being square does not guarantee that the inverse exists.

To find the inverse of a matrix such that , we apply the formula where . Notice that if the determinant of matrix is equal to zero, the inverse cannot exist. If the determinant is not zero, matrix will have an inverse. We then call matrix invertible or nonsingular. The properties of inverse matrices we will consider in this lesson will apply to all invertible matrices.

Let’s use the definition for an inverse matrix to derive some of the key properties of inverse matrices.

### Example 1: Identifying an Equivalent Expression for Matrices Using the Properties of Inverse Matrices

If is a matrix, which of the following is equal to ?

Since exists, must be a square matrix. Let’s imagine is a matrix such that

Based on the definition of the inverse for a matrix,

If we square the inverse of , we have

Next, we want to calculate

Taking the inverse of the square of matrix , we have

Note that the property of determinants that allows us to calculate the determinant of to be .

Since the expression for the inverse of the square of matrix is the same as the expression for the square of the inverse of matrix , we have shown that, for an invertible matrix ,

In the previous example, we demonstrated that . This can be generalized for higher powers of the inverse matrix such that, for any invertible matrix where ,

In our next example, we will consider the relationship between the transpose of an inverse and the inverse of a transpose.

### Example 2: Identifying an Equivalent Expression for Matrices Using the Properties of Inverse Matrices

If is a matrix, which of the following is equal to ?

Recall that the capital written in superscript text is the notation for the transpose of a matrix. This means that we switch the rows with the columns. When we transpose a matrix, the values along the diagonal do not change.

Since exists, must be a square matrix. Let’s imagine is a matrix such that

Based on the definition of the inverse for a matrix,

If we take the transpose of the inverse of matrix , we have

Note that since the fraction can be distributed across the matrix, taking the transpose does not affect it and so we can leave it on the outside of the matrix.

Next, we want to calculate the transpose of matrix , which gives

Taking the inverse of the transpose of matrix would then yield

We have just demonstrated that, for a invertible matrix,

In the previous example, we demonstrated that for a matrix. This can be generalized for all invertible matrices :

In the next example, we will explore what happens when we find the inverse of the inverse of a matrix.

### Example 3: Using Properties of Inverse Matrices to Solve a Problem

Consider the matrix . Find .

To find the inverse of the inverse of matrix , we first want to find the inverse of matrix . For a matrix of the form the inverse will be equal to

Therefore, in this example, the inverse will be

If we then found the inverse of , we would take which simplifies to

Then, we multiply by the scalar :

We have just shown that the inverse of the inverse of a matrix is the original matrix. However, it is not necessary to do all of these calculations to solve this problem because this property is true for all invertible matrices:

Once we show that matrix has a nonzero determinant, the properties of inverse matrices tell us that the inverse of the inverse will be the original matrix,

In the previous example, we demonstrated that for a matrix. This can be generalized for all invertible matrices:

In the next example, we will see how we can use the properties of inverse matrices and the identity matrix to simplify problem solving.

### Example 4: Using Properties of Inverse Matrices to Solve a Problem

1. Given the matrices and , where and , find .
2. Without doing further calculations, find .

Part 1

The first part of this question has asked us to multiply matrix by matrix . As both of these matrices are square matrices, they can be multiplied together, remembering that matrix multiplication is not commutative. To multiply matrices, we find the dot product of the rows in the first matrix and the columns in the second:

When we evaluate each of these expressions, we find

Part 2

The second part of the question asked us to find the of inverse of matrix without doing any further calculations. When we multiplied the matrices and , the resultant matrix was the identity matrix. We recall that any square matrix (with a nonzero determinant) has an inverse such that . Since the product of matrix and matrix was the identity matrix, matrix must be the inverse of matrix .

Therefore,

There is one further property that we can use: the relationship between the product of two matrices and their inverse.

For a pair of invertible matrices and ,

In our final example, we will look at how to apply this connection between the inverse of the product of two matrices and the product of their respective inverses.

### Example 5: Using Properties of Inverse Matrices to Solve a Problem

Given that determine .

We recall, given a pair of invertible matrices and , . This means that we can rewrite our first equation as

We now need some strategy to “cancel out” the from both sides of this equation. To do that, we use a key property of the identity matrix. We know that . This means that we can multiply both sides of this equation by matrix that we were given, remembering that matrix multiplication is not commutative; we must multiply in the correct order:

From there, we follow the rules for multiplying matrices and find that

On the left side of this equation, we see that we have multiplied the inverse of matrix by matrix . We know that this equals the identity matrix. So, we can rewrite the left side of the equation to be

Remembering that multiplying any matrix by the identity matrix produces itself, we can say

The final step will be to multiply through by the scalar to produce

Let’s finish by recapping the key properties we use when working with inverse matrices.

### Key Points

• The product of a matrix and its inverse is the identity matrix:
• The inverse of the inverse of a matrix is the matrix itself:
• The inverse of a matrix to the power is equal to the power of the inverse of the matrix:
• The inverse of the product of matrix and matrix is equal to the product of the inverse of matrix and the inverse of matrix :
• The transpose of the inverse of a matrix is equal to the inverse of the transpose of a matrix: