We introduce elementary matrices and demonstrate how multiplication of a matrix by an elementary matrix is equivalent to to performing an elementary row operation on the matrix.

MAT-0060: Elementary Matrices

Definition and Examples

Consider the matrices The two matrices have something in common. Can you figure out what it is? (The answer will be given later in the problem.)

Let’s compute and .

Observe that multiplying by on the left results in multiplying the second row of by , while multiplying by on the left results in multiplying the third row of by .

Now we need to return to the question of what and have in common. Both matrices were obtained from the identity matrix by multiplying one row of the identity by a non-zero constant. Matrices and were obtained from by multiplying one row of by and respectively. Multiplying by (or ) on the left affects in the same way.

Matrix does not have to be a square matrix. Try finding and for

Observe that and have the same effect on as they did on .

In general, if a square matrix is obtained from the identity matrix by multiplying row of by a non-zero constant , then multiplying an appropriately sized matrix on the left by results in row of being multiplied by .

Recall that multiplication of a row of a matrix by a non-zero constant is one of three elementary row operations. Applying such an elementary row operation to in order to produce , results in applying the same elementary row operation to when is multiplied by on the left.

Consider the matrices As in the previous Exploration, the two matrices have something in common. Both and were obtained from the identity matrix by adding a multiple of one row to another row. Can you guess what will happen if we multiply a matrix by and on the left?

Let’s compute and .

As you had probably guessed, multiplication by resulted in the third row of being added to the first, and multiplication by produced a matrix by adding times the first row to the second row of . The elementary row operations performed on mimic the elementary row operations performed on in order to obtain and .

In general, if a square matrix is obtained from the identity matrix by adding times row of to row , then multiplying an appropriately sized matrix on the left by results in times row of being added to row of .

Recall that adding a scalar multiple of one row to another row of a matrix is one of three elementary row operations. Applying such an elementary row operation to in order to produce , results in applying the same elementary row operation to when is multiplied by on the left.

The matrices above are special because when we multiply them by any appropriately sized matrix , we are performing row operations on . Can you construct a matrix such that is the same as except that its first and third rows are switched?

The matrices of Explorations init:elementarymat2, init:elementarymat1 and init:elementarymat3 are known as elementary matrices because they perform elementary row operations on appropriately sized matrices.

Suppose is an matrix. If is an elementary matrix formed by performing a certain row operation on the identity matrix, then multiplying any matrix on the left by is equivalent to performing that same row operation on . As there are three types of elementary row operations, there are three types of elementary matrices.

Elementary matrices give us a new way of looking at Gauss-Jordan elimination. Suppose it takes elementary row operations to transform into , its reduced row-echelon form. Then we can represent this reduced row-echelon form as where each is the elementary matrix corresponding to the th row operation performed on .

Inverses of Elementary Matrices

It is easy to see that any elementary matrix is invertible, because if is formed by applying a certain row operation to the identity matrix , then there is a single row operation that may be applied to to get back. For example, in Exploration init:elementarymat1, is formed by adding times the first row of the identity to the second row of the identity. It follows that should be the matrix formed by adding times the first row of the identity to the second row of the identity, i.e.

And indeed we can check and also .

As part of the Practice Problem set you are asked to find the inverse of each of the other elementary matrices in Explorations init:elementarymat2, init:elementarymat1 and init:elementarymat3. Once we have accounted for each of the three types of elementary matrices, we will have proven the following theorem.

Proof
Suppose is obtained from by switching rows and . To find the inverse of , we need to find a matrix such that . To get from back to , rows and of must be switched. This can be accomplished by multiplying by itself on the left. So, is its own inverse.

We can use the same line of reasoning to show that the other two types of elementary matrices are also invertible, and their inverses are also elementary matrices. The details are left to the reader.

Elementary Matrices and Nonsingular Matrices

Recall that a square matrix is called nonsingular provided that .

We will prove equivalence of the three statements by showing that

item:anonsingular2item:aproductofelemmatricesitem:ainvertibleitem:anonsingular2

Proof of item:anonsingular2item:aproductofelemmatrices
Suppose . Then can be carried to the identity by elementary row operations. So, there exist elementary matrices such that By Theorem th:elemmatricesinvertible, elementary matrices are invertible and their inverses are also elementary matrices. Thus, we can write as a product of elementary matrices as follows:

Proof of item:aproductofelemmatricesitem:ainvertible
Suppose , where are elementary matrices. In Theorem item:inverseofproduct of MAT-0050 we proved that . By repeated applications of this theorem we have We conclude that is invertible.

Proof of item:ainvertibleitem:anonsingular2
See the corollary to Theorem th:matrixinverse of MAT-0050.

Practice Problems

For each elementary matrix below, determine the elementary row operation that results from multiplying a matrix by on the left. Write down without going through the row-reduction procedure.
Think of an elementary row operation that would undo the row operation caused by .
Answer:
Answer:
Answer:
Find the inverse of each of the following elementary matrices from Explorations init:elementarymat2, init:elementarymat1 and init:elementarymat3.
Finish the proof of Theorem th:elemmatricesinvertible.
Express as a product of elementary matrices.
Row-reduce to find . Record the elementary row operations as you perform row reduction. You will be able to conclude that . Find the inverse of each and multiply by the inverses on the left.
Is this representation unique? Prove your claim.
In Explorations init:elementarymat2, init:elementarymat1 and init:elementarymat3 we performed elementary row operations on by multiplying by elementary matrices on the left. Compute and . Summarize your findings.

Answer:

If possible, express as a product of elementary matrices.