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Mathematical Expression Editor

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?

An elementary matrix is a square matrix formed by applying a single elementary row
operation to the identity matrix.

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.

Elementary matrices are invertible, and the inverse of an elementary matrix is an
elementary matrix.

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 .

The following statements are equivalent for an matrix .

(a)

is nonsingular

(b)

is a product of elementary matrices

(c)

is invertible

We will prove equivalence of the three statements by showing that

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:

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.

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 .

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.