In this explainer, we will learn how to find the inverse of matrices using the adjoint method.
Let us begin by recalling how to define the inverse of a matrix.
Definition: Inverse of a 2 × 2 Matrix
Let be a matrix. The inverse of (denoted by ) is a matrix that satisfies where is the identity matrix. If such a matrix exists, we say that matrix is invertible.
Furthermore, it is in fact possible to derive an exact formula for the inverse, which is as follows.
Formula: Inverse of a 2 × 2 Matrix
Let such that , where is the determinant of . Then the inverse of is given by
If , matrix is not invertible.
It stands to reason that if there is such a thing as an inverse for matrices, the same must be true for higher-order matrices. As expected, the definition of an inverse of a matrix can indeed be extended to include matrices of any order, as follows.
Definition: Inverse of a Matrix
Let be an matrix. The inverse of (denoted by ) is an matrix that satisfies where is the identity matrix. If such a matrix exists, we say that matrix is invertible.
Having said that this extension is possible, it is easier said than done to derive formulas for such matrices or to know if they even exist in the first place. In the case, we note that the inverse is obtained by manipulating the entries of the matrix and dividing by the determinant, provided it is not equal to zero. We might wonder whether a similar approach exists for higher-dimensional cases.
As we will find out in this explainer, there does exist a formula for the matrix inverse that generalizes the case. In particular, finding the determinant and the steps involved in doing so are a key component of the process. As the primary focus of this explainer is matrices, we will be reviewing the method for calculating the determinant of a matrix using cofactor expansion. However, since the full method will only be needed later, let us begin with the first step in the process, which is the calculation of minors and cofactors.
Definition: Minors and Cofactors
Let be a matrix of order . Then, the minor of element (denoted by ) is the determinant of the matrix obtained after removing row and column from .
Then, the cofactor of element (denoted by ) is equal to where is the minor of element .
So far, we have only seen cofactors used to calculate the determinant of a matrix. However, as it turns out, they are also essential in finding the inverse of a matrix using the adjoint method.
Before we properly explain the adjoint method for finding the inverse, we need to define cofactor matrices.
Definition: Cofactor Matrix
The cofactor matrix of a square matrix is defined by where each is the cofactor of entry of .
The above definition applies to any square matrix, but in the case, this is
In other words, each entry of the matrix is the cofactor of the corresponding entry in the original matrix.
Although this is a simple formula to state, it can be tricky to calculate in practice, since we have to find the minor and its cofactor for each entry of the cofactor matrix. Let us practice this with an example.
Example 1: Finding the Cofactor Matrix of a 3 × 3 Matrix
Find the cofactor matrix of
Answer
Recall that the cofactor matrix is the matrix obtained by finding the cofactor of each corresponding entry of a matrix. Let the given matrix be .
To find the cofactors, we first have to find the minors. To do this, for each entry of the matrix, we remove the row and column that it belongs to and take the determinant of the resulting matrix. For instance, the minor of entry can be found as follows:
Repeating this process, we get nine different determinants, where each time we have removed the row and column that the corresponding entry belongs to. For the first row, we have
For the second row, we have
For the final row, we have
Now, we need to find the cofactors. Recall that the cofactors can be obtained from the corresponding minors by multiplying them by 1 or , according to their position in the following matrix:
For instance, is in position , which has a negative sign, which shows us that . In summary, we have
Finally, we can put these into a matrix to form the cofactor matrix. This gives us
Before we introduce the formula for the inverse of a matrix, we must introduce one final concept: the adjoint matrix. We define it as follows.
Definition: Adjoint Matrices
The adjoint of (also known as the adjugate) is the transpose of the cofactor matrix ; that is,
As we can see, once we have calculated the cofactor matrix, it is a simple procedure to obtain the adjoint matrix, since we just have to take the transpose. Let us now consider an example where we have to calculate the adjoint matrix.
Example 2: Finding the Adjoint Matrix of a 3 × 3 Matrix
Find the adjoint matrix of the matrix
Answer
Recall that the adjoint matrix is the transpose of the cofactor matrix, which is a matrix where each entry is a cofactor of the corresponding entry. Let the given matrix be .
To find the cofactors of , we first have to find the minors, which we can do by taking each entry of the matrix one by one, removing the rows and columns that they belong to, and taking the determinants of the resulting matrices. Let us demonstrate this for the minor of :
Let us continue this approach to find each of the nine minors, where each time we remove the row and column that the corresponding entry belongs to. For the first row, we have
For the second row, we have
For the final row, we have
Now, we can use the minors to find the cofactors. Recall that the cofactors can be obtained from the corresponding minors by multiplying them by 1 or , according to their position in the following matrix:
In summary, we have
We can now form the cofactor matrix, by putting the entries into the following matrix:
Finally, to find the adjoint matrix, we transpose . This means we must rewrite each of the rows as a column of the new matrix. This gives us
Before we proceed to define the formula for the inverse of a matrix, let us first recall the method for finding the determinant of a matrix using cofactor expansion.
Definition: Determinants of 3 × 3 Matrices (Cofactor Expansion)
For any fixed , 2, or 3, the determinant of is equal to where each is the cofactor of element . This is known as the cofactor expansion (or Laplace expansion) along row . Alternatively, for any fixed , 2, or 3, we have
This is the cofactor expansion along column .
With this definition and the prior definitions of the cofactor and adjoint matrices, we are now in a position to write the formula for the inverse of a matrix.
Formula: Inverse of a Matrix
If is an invertible matrix, then its inverse is where is the adjoint of and is the determinant of .
We note that this formula applies to square matrices of any order, although we will only use it to find inverses here. It is however possible to show that this formula works for the case too. To see this, let us consider the general matrix:
To find , we first calculate the cofactor matrix by finding each of the minors. We note that, in this case, finding the minor for each entry is rather trivial, since removing a row and a column from a matrix results in a matrix, which is just a number. So, the four minors are just the entries in the opposite corners, as shown:
To find the cofactor matrix, we reverse the signs of and , to get the following matrix:
Finally, we obtain by taking the transpose of , to get
Putting this into the above formula, we have
Since this is the same as the formula for the inverse we already had, we can therefore see that the two formulas are consistent.
It is important to note that is invertible if and only if the determinant is nonzero. This means that whenever we have to find the inverse of a matrix, we should always start by computing the determinant. If it is nonzero, then we can proceed; otherwise, we must conclude that the matrix is singular (i.e., has no inverse).
Let us test our ability to find the inverse of matrices.
Example 3: Checking Whether a 3 × 3 Matrix Is Singular and Finding Its Inverse If Possible
Determine whether the matrix has an inverse by finding whether the determinant is nonzero. If the determinant is nonzero, find the inverse using the inverse formula involving the cofactor matrix.
Answer
The first part of the question requires us to check whether the determinant is nonzero. Naturally, this can be done by computing the determinant directly. However, we can also use the properties of determinants to help us calculate the determinant more easily and in some cases show that it must be zero.
Recall that if we add a scalar multiple of one row to another row, the value of the determinant does not change. Thus, let us add times row 1 to row 3. We have chosen to do this because it will result in the bottom-left entry becoming zero. This gives us
Now, ordinarily, we would continue computing the determinant by using cofactor expansion on the first column. However, at this stage, it is important to realize that the second and third rows are both . Let us recall another property of determinants, which is that if two rows of a matrix are equal, the determinant is zero. Since that is the case here, we no longer need to proceed with the rest of the calculation because the property tells us that it is zero.
Thus, there is no inverse because the determinant equals zero.
In the previous example, the matrix was singular, so we did not have to go through all the steps of calculating the inverse. This shows that it is always very useful to find the determinant before going to the trouble of finding the adjoint matrix. Let us consider another example where we have to try and find the inverse of a matrix.
Example 4: Checking Whether a 3 × 3 Matrix Is Singular and Finding Its Inverse If Possible
Consider the matrix
- Determine whether the matrix has an inverse by finding whether the determinant is nonzero.
- If the determinant is nonzero, find the inverse using the formula for the inverse that involves the cofactor matrix.
Answer
Part 1
The first part of the question asks us to find whether the determinant is nonzero, so let us calculate the determinant. We note that the second column has two zeros, which means it is already in the optimal form for applying the method of cofactor expansion. Recall that the cofactor expansion along column is
Let the given matrix be . Now, if we take , the calculation simplifies to
Thus, we only need to find . To calculate it, we first find the minor of by removing the row and column that is in, and taking the determinant. This gives us
Next, we can find the cofactor by applying the definition
Since , , and , we have
Thus, we have . Since the determinant is nonzero, the matrix has an inverse.
Part 2
Recall that the inverse is given by the formula
We have already calculated , so let us now calculate the adjoint matrix, which is the transpose of the cofactor matrix.
To find the cofactor matrix, we have to first find the minor in each position of the matrix. For the first row, these minors are
Continuing with the second and third rows, where we have left out the intermediate steps for brevity, we have
Now, we can use the minors to find the cofactors. Recall that the cofactors can be obtained from the corresponding minors by multiplying them by 1 or , according to their position in the following matrix:
Doing this for each cofactor and putting them into a matrix, we have
Now, we can find the adjoint by taking the transpose of this matrix. We do this by rewriting each of the rows of as a column of . This gives us
Finally, we use the formula for the inverse, :
At this point, we should be somewhat familiar with the method of finding the inverse of a matrix, but we have yet to do anything interesting with the resulting inverse matrix. One key use of the inverse matrix is in solving matrix equations. Consider an equation of the form where and are given matrices and is an unknown matrix. If is not singular, then the inverse exists, and we can multiply both sides of the equation on the left by to get
Thus, if we find the inverse of , we can use it to find the unknown matrix as shown. Let us see a full example of this.
Example 5: Solving a Matrix Equation Using the Inverse of a Matrix
Suppose that , where and is a matrix.
- Calculate the inverse of .
- And use it to find .
Answer
Part 1
Let us begin by calculating the inverse of using the adjoint method. Recall that we have
Thus, let us first calculate followed by and use them to find the inverse. Now, we can find the inverse of by using cofactor expansion on column 2 (since one entry is already zero). In other words, we use the formula where and are the minors of and respectively. We can find the minors by removing the corresponding rows and columns from the matrix and taking the determinant of the result. Doing this for both minors, we get
Thus, . This indeed confirms that the inverse exists (although this is assumed since the question would have not have a unique solution otherwise). Now, we need to find the adjoint of . We do this by first computing the minors and then the cofactor matrix. The minors of are as follows:
We can now construct the cofactor matrix, which we can do by multiplying each minor by 1 or , according to their position in the following matrix:
Doing this gets us the following cofactor matrix:
Lastly, we can get the adjoint matrix by taking the transpose of the above matrix. This gives us
Since the determinant is 1, we have , which means that the inverse is equal to the above matrix:
Part 2
Now we have found , we can find in the equation . If we multiply on the left of both sides of the equation by , then we find that
Thus, we just need to perform the matrix multiplication of and . Let us demonstrate this for the first entry:
where we have done the calculation . Repeating this for the remaining entries, we get
For our final example, we will consider a situation where we have to find the inverse of a matrix where the entries are variable quantities.
Example 6: Finding the Inverse of a Matrix with Variable Entries Using the Adjoint Method
Find the inverse of the matrix
Answer
In this question, it is assumed that the given matrix is invertible, so we do not strictly need to check whether the determinant is nonzero. Nevertheless, it is still necessary to calculate the determinant in order to find the inverse, since it is given by where is the determinant and is the adjoint matrix (i.e., the transpose of the cofactor matrix). Let the given matrix be . Recall that the determinant can be calculated using cofactor expansion along row :
Ordinarily, it is useful to manipulate the matrix so that two out of the three entries in a single row or column are 0. This means we only need to find a single cofactor. However, since we will have to calculate all the cofactors anyway for the cofactor matrix, let us just proceed to use cofactor expansion on the first row.
In order to find the cofactors of row 1, we first have to find the minors. We can do this for each entry by removing the row and column that that entry belongs to and taking the determinant of the resulting matrix. For instance, the minor for is
where we have used the Pythagorean trigonometric identity, . Continuing with the other two entries, we get
From here, recall that . Thus, we have
Now, the formula for cofactor expansion along row 1 is
Using the fact that , , and , this means that
As expected, the determinant is nonzero. We now need to find the rest of the cofactors by calculating the corresponding minors. For the second and third rows, the minors are
Let us now calculate the cofactors, which we can obtain by multiplying each corresponding minor by 1 or , according to their position in the following matrix:
Computing these cofactors and putting them into a matrix, we have
Next, the adjoint matrix can be obtained by taking the transpose of this matrix, which means rewriting the rows as columns. Doing this gives us
Finally, we use the formula to get
Let us summarize the key points we have learned during this explainer.
Key Points
- Let be a matrix of order . Then, the minor of element (denoted by ) is the determinant
of the matrix obtained after removing row and column from
.
Then, the cofactor of element (denoted by ) is equal to where is the minor of element . - If is a square matrix, then its cofactor matrix is defined by where each is the cofactor of entry of .
- The adjoint of (also known as the adjugate) is the transpose of the cofactor matrix ; that is,
- If is an invertible matrix (i.e., ), then its inverse is where is the determinant of .
- Since a matrix is only invertible if the determinant is nonzero, it is important to check whether this is true before attempting to find the inverse using the adjoint method. We can make this process easier by using properties of determinants.
- Given a matrix equation of the form , we can find by multiplying on the left by as follows:
