In this explainer, we will learn how to check whether a matrix has an inverse and then find its inverse, if possible.
Recall that a matrix that has the same number of columns and rows is called a square matrix. By the definition of matrix products, any two matrices can be multiplied together, and we can make the following definition.
Definition: The Inverse of an 𝑛 × 𝑛 Matrix
The inverse of an matrix is a matrix such that where is the identity matrix.
- We call the inverse of because it is unique and we denote it by . In particular, if we know that , then is just .
- The definition says what is expected of an inverse, but it does not say that such an inverse exists. For example, consider , the zero matrix. Then for any matrix , we have Therefore, we can never get as the product with any because the zero matrix has no inverse.
- On the other hand, the identity matrix is its own inverse because
One approach to finding the inverse of is to try and solve some equations. Suppose that . Then we must solve for in the system
In other words,
This is a system of 4 equations in the 4 unknowns, not as straightforward as we would like.
A simplification comes from the observation that given , if we look at the matrix then we find products
This gives a diagonal matrix with the determinant of —the number —on the main diagonal.
We conclude two things from this fact:
- If , then we have found a matrix whose product with is the zero matrix. It is not hard to see that it is, therefore, impossible for to have an inverse.
- If , then the matrix is the inverse of .
The matrix is called the adjugate of and we get it by two steps:
- Swap the main diagonal entries: .
- Negate the off-diagonal entries: , .
For example, with we have and adjugate and, therefore, inverse
- The inverse of an matrix is an matrix such that , where is the identity matrix. We call this matrix .
- For matrices, the matrix has an inverse exactly when its determinant is not zero.
- When the matrix is invertible (has an inverse), then it is given as times the adjugate matrix .
Example 1: Identifying Whether a Given Matrix Is Invertible Using the Determinant
Is the following matrix invertible?
We know that this matrix, which we will call , is invertible if, and only if, it has a nonzero determinant. We compute
Therefore, this matrix is not invertible.
Example 2: Identifying Whether Two Matrices Are Inverses of One Another
Are the matrices multiplicative inverses of each other?
Of course, “multiplicative” is for emphasis. Usually, the word “inverse” is enough. To decide whether they are inverses or not, we multiply the matrices together (in either order). We will find which is not the identity matrix.
These matrices are not inverses of each other.
Example 3: Finding the Inverse of a Matrix
Find the multiplicative inverse of the matrix , if possible.
We determine whether an inverse exists at all by calculating the determinant:
It is not zero, so we can proceed to find the inverse: