3.1 Elementary Matrix Operations and Elementary Matrices – Linear Algebra, 5th Edition

3.1 Elementary Matrix Operations and Elementary Matrices

In this section, we define the elementary operations that are used throughout the chapter. In subsequent sections, we use these operations to obtain simple computational methods for determining the rank of a linear transformation and the solution of a system of linear equations. There are two types of elementary matrix operations—row operations and column operations. As we will see, the row operations are more useful. They arise from the three operations that can be used to eliminate variables in a system of linear equations.

Definitions.

Let A be an m×n matrix. Any one of the following three operations on the rows [columns] of A is called an elementary row [column] operation:

  1. (1) interchanging any two rows [columns] of A;

  2. (2) multiplying any row [column] of A by a nonzero scalar;

  3. (3) adding any scalar multiple of a row [column] of A to another row [column].

Any of these three operations is called an elementary operation. Elementary operations are of type 1, type 2, or type 3 depending on whether they are obtained by (1), (2), or (3).

Example 1

Let

A=(123421134012).

Interchanging the second row of A with the first row is an example of an elementary row operation of type 1. The resulting matrix is

B=(211312344012).

Multiplying the second column of A by 3 is an example of an elementary column operation of type 2. The resulting matrix is

C=(163423134012).

Adding 4 times the third row of A to the first row is an example of an elementary row operation of type 3. In this case, the resulting matrix is

M=(17271221134012).

Notice that if a matrix Q can be obtained from a matrix P by means of an elementary row operation, then P can be obtained from Q by an elementary row operation of the same type. (See Exercise 8.) So, in Example 1, A can be obtained from M by adding 4 times the third row of M to the first row of M.

Definition.

An n×n elementary matrix is a matrix obtained by performing an elementary operation on In. The elementary matrix is said to be of type 1, 2, or 3 according to whether the elementary operation performed on In is a type 1, 2, or 3 operation, respectively.

For example, interchanging the first two rows of I3 produces the elementary matrix

E=(010100001).

Note that E can also be obtained by interchanging the first two columns of I3. In fact, any elementary matrix can be obtained in at least two ways— either by performing an elementary row operation on In or by performing an elementary column operation on In. (See Exercise 4.) Similarly,

(102010001)

is an elementary matrix since it can be obtained from I3 by an elementary column operation of type 3 (adding 2 times the first column of I3 to the third column) or by an elementary row operation of type 3 (adding 2 times the third row to the first row).

Our first theorem shows that performing an elementary row operation on a matrix is equivalent to multiplying the matrix by an elementary matrix.

Theorem 3.1.

Let AMm×n(F), and suppose that B is obtained from A by performing an elementary row [column] operation. Then there exists an m×m [n×n] elementary matrix E such that B=EA [B=AE]. In fact, E is obtained from Im [In] by performing the same elementary row [column] operation as that which was performed on A to obtain B. Conversely, if E is an elementary m×m [n×n] matrix, then EA [AE] is the matrix obtained from A by performing the same elementary row [column] operation as that which produces E from Im [In].

The proof, which we omit, requires verifying Theorem 3.1 for each type of elementary row operation. The proof for column operations can then be obtained by using the matrix transpose to transform a column operation into a row operation. The details are left as an exercise. (See Exercise 7.)

The next example illustrates the use of the theorem.

Example 2

Consider the matrices A and B in Example 1. In this case, B is obtained from A by interchanging the first two rows of A. Performing this same operation on I3, we obtain the elementary matrix

E=(010100001).

Note that EA=B.

In the second part of Example 1, C is obtained from A by multiplying the second column of A by 3. Performing this same operation on I4, we obtain the elementary matrix

E=(1000030000100001).

Observe that AE=C.

It is a useful fact that the inverse of an elementary matrix is also an elementary matrix.

Theorem 3.2.

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

Proof.

Let E be an elementary n×n matrix. Then E can be obtained by an elementary row operation on In. By reversing the steps used to transform In into E, we can transform E back into In. The result is that In can be obtained from E by an elementary row operation of the same type. By Theorem 3.1, there is an elementary matrix E¯ such that E¯E=In. Therefore, by Exercise 10 of Section 2.4, E is invertible and E1=E¯.

Exercises

  1. Label the following statements as true or false.

    1. (a) An elementary matrix is always square.

    2. (b) The only entries of an elementary matrix are zeros and ones.

    3. (c) The n×n identity matrix is an elementary matrix.

    4. (d) The product of two n×n elementary matrices is an elementary matrix.

    5. (e) The inverse of an elementary matrix is an elementary matrix.

    6. (f) The sum of two n×n elementary matrices is an elementary matrix.

    7. (g) The transpose of an elementary matrix is an elementary matrix.

    8. (h) If B is a matrix that can be obtained by performing an elementary row operation on a matrix A, then B can also be obtained by performing an elementary column operation on A.

    9. (i) If B is a matrix that can be obtained by performing an elementary row operation on a matrix A, then A can be obtained by performing an elementary row operation on B.

  2. Let

    A=(123101111), B=(103121131),  and C=(103022131).

    Find an elementary operation that transforms A into B and an elementary operation that transforms B into C. By means of several additional operations, transform C into I3.

  3. Use the proof of Theorem 3.2 to obtain the inverse of each of the following elementary matrices.

    1. (a) (001010100)

    2. (b) (100030001)

    3. (c) (100010201)

  4. Prove the assertion made on page 149: Any elementary n×n matrix can be obtained in at least two ways—either by performing an elementary row operation on In or by performing an elementary column operation on In.

  5. Prove that E is an elementary matrix if and only if Et is.

  6. Let A be an m×n matrix. Prove that if B can be obtained from A by an elementary row [column] operation, then Bt can be obtained from At by the corresponding elementary column [row] operation.

  7. Prove that if a matrix Q can be obtained from a matrix P by an elementary row operation, then P can be obtained from Q by an elementary row operation of the same type. Hint: Treat each type of elementary row operation separately.

  8. Prove that any elementary row [column] operation of type 1 can be obtained by a succession of three elementary row [column] operations of type 3 followed by one elementary row [column] operation of type 2. Visit goo.gl/oNJBFz for a solution.

  9. Prove that any elementary row [column] operation of type 2 can be obtained by dividing some row [column] by a nonzero scalar.

  10. Prove that any elementary row [column] operation of type 3 can be obtained by subtracting a multiple of some row [column] from another row [column].

  11. Let A be an m×n matrix. Prove that there exists a sequence of elementary row operations of types 1 and 3 that transforms A into an upper triangular matrix.