The Inverse of a Matrix
Definition and Properties of Matrix Inverses
Consider the equation . It takes little time to recognize that the solution to this equation is . In fact, the solution is so obvious that we do not think about the algebraic steps necessary to find it. Let’s take a look at these steps in detail. This process utilizes many properties of real-number multiplication. In particular, we make use of existence of multiplicative inverses. Every non-zero real number has a multiplicative inverse with the property that . We say that is the multiplicative identity because .
Given a matrix equation , we would like to follow a process similar to the one above to solve this matrix equation for .
Observe that the role of the multiplicative identity for square matrices is filled by because . Given an matrix , a multiplicative inverse of would have to be some matrix such that Assuming that such an inverse exists, this is what the process of solving the equation would look like:
Let be an matrix. An matrix is called an inverse of if
where is an identity matrix. If such an inverse matrix exists, we say that is
invertible. If an inverse does not exist, we say that is not invertible.
It follows directly from the way the definition is stated that if is an inverse of , then
is an inverse of . We say that and are inverses of each other.
The following theorem shows that matrix inverses are unique.
- Proof
- Because is an inverse of , we have: Suppose there exists another matrix such that Then
Now that we know that a matrix cannot have more than one inverse, we can safely refer to the inverse of as .
We now prove several useful properties of matrix inverses.
We will prove item:inverseofproduct. The remaining properties are left as exercises.
- Proof of Property item:inverseofproduct:
- We will check to see if is the inverse of . Thus is invertible and .
Computing the Inverse
We now turn to the question of how to find the inverse of a square matrix, or determine that the inverse does not exist.
Given a square matrix , we are looking for a square matrix such that We will start by attempting to satisfy . Let be the columns of , then where each is a standard unit vector of . This gives us a system of equations for each . If each has a unique solution, then finding these solutions will give us the columns of the desired matrix .
First, suppose that , then we can use elementary row operations to carry each to its reduced row-echelon form. Observe that the row operations that carry to will be the same for each . We can, therefore, combine the process of solving systems of equations into a single process Each is a unique solution of , and we conclude that is a solution to .
By Problem prob:elemrowopsreverse, we can reverse the elementary row operations to obtain But the same row operations would also give us We conclude that , and .
Next, suppose that . Then must contain a row of zeros. Because one of the rows of was completely wiped out by elementary row operations, one of the rows of must be a linear combination of the other rows. Suppose row is a linear combination of the other rows. Then row can be carried to a row of zeros. But then the system is inconsistent. This is because has a as the entry and zeros everywhere else. The in the spot will not be affected by elementary row operations, and the row will eventually look like this This shows that a matrix such that does not exist, and does not have an inverse.
We have just proved the following theorem.
Row-reduction Method for Computing the Inverse of a Matrix Let be a square
matrix. If it is possible to use elementary row operations to carry the augmented
matrix to , then . If such a reduction is not possible, then does not have an
inverse.
Recall that a square matrix whose reduced row-echelon form is the identity matrix is
called nonsingular. (Definition def:nonsingularmatrix) According to Corollary cor:rrefI, a matrix is invertible if and
only if it is nonsingular. For this reason many linear algebra texts use the terms
invertible and nonsingular as synonyms.
Inverse of a Matrix
We will conclude this section by discussing the inverse of a nonsingular matrix. Let be a nonsingular matrix. We can find by using the row reduction method described above, that is, by computing the reduced row-echelon form of . Row reduction yields the following:
Note that the denominator of each term in the inverse matrix is the same. Factoring it out, gives us the following formula for .
Clearly, the expression for is defined, if and only if . So, what happens when ? In Practice Problem prob:inverseformula you will be asked to fill in the steps of the row reduction procedure that produces this formula, and show that if then does not have an inverse.
Practice Problems
Use the row-reduction method for computing matrix inverses to explain why the
given matrix does not have an inverse.
Use the following invertible matrices to illustrate Properties item:inverseofinverse, item:inverseofproduct, item:inversekA and item:inversetranspose of Theorem
th:invprop.
Use the row-reduction method to prove Formula form:detinverse for a nonsingular matrix. Show
that if then does not have an inverse.
For each matrix below refer to Formula form:detinverse to find the value of for which the matrix is
not invertible.
Suppose is a symmetric, invertible matrix. Does it follow that is symmetric? What
if we change “symmetric” to “skew symmetric”? (See Definition def:symmetricandskewsymmetric.)