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Elementary matrix operations play an important role in many matrix algebra applications, such as finding the inverse of a matrix and solving simultaneous linear equations.

There are three kinds of elementary matrix operations.

- Interchange two rows (or columns).
- Multiply each element in a row (or column) by a non-zero number.
- Multiply a row (or column) by a non-zero number and add the result to another row (or column).

When these operations are performed on rows, they are called **elementary row operations**; and when they are performed on columns, they are called **elementary column operations**.

In many references, you will encounter a compact notation to describe elementary operations. That notation is shown below.

Each type of elementary operation may be performed by matrix multiplication, using square matrices called**elementary operators**.

For example, suppose you want to interchange rows 1 and 2 of Matrix **A**. To accomplish this, you could premultiply **A** by **E** to produce **B**, as shown below.

Here, **E** is an elementary operator. It operates on **A** to produce the desired interchanged rows in **B**. What we would like to know, of course, is how to find **E**. Read on.

To perform an elementary row operation on a **A**, an *r* x *c* matrix, take the following steps.

- To find
**E**, the**elementary row operator**, apply the operation to an*r*x*r*identity matrix. - To carry out the elementary row operation, premultiply
**A**by**E**.

We illustrate this process below for each of the three types of elementary row operations.

**Interchange two rows**. Suppose we want to interchange the second and third rows of**A**, a 3 x 2 matrix. To create the elementary row operator**E**, we interchange the second and third rows of the identity matrix**I**._{3}

Then, to interchange the second and third rows of**A**, we premultiply**A**by**E**, as shown below.

**Multiply a row by a number**. Suppose we want to multiply each element in the second row of Matrix **A** by 7. Assume **A** is a 2 x 3 matrix. To create the elementary row operator **E**, we multiply each element in the second row of the identity matrix **I _{2}** by 7.

Then, to multiply each element in the second row of **A** by 7, we premultiply **A** by **E**.

**Multiply a row and add it to another row**. Assume**A**is a 2 x 2 matrix. Suppose we want to multiply each element in the first row of**A**by 3; and we want to add that result to the second row of**A**. For this operation, creating the elementary row operator is a two-step process. First, we multiply each element in the first row of the identity matrix**I**by 3. Next, we add the result of that multiplication to the second row of_{2}**I**to produce_{2}**E**

Then, to multiply each element in the first row of**A**by 3 and add that result to the second row, we premultiply**A**by**E**.## How to Perform Elementary Column Operations

To perform an elementary column operation on

**A**, an*r*x*c*matrix, take the following steps.

- To find
**E**, the**elementary column operator**, apply the operation to an*c*x*c*identity matrix. - To carry out the elementary column operation, postmultiply
**A**by**E**.

Let's work through an elementary column operation to illustrate the process. For example, suppose we want to interchange the first and second columns of

**A**, a 3 x 2 matrix. To create the elementary column operator**E**, we interchange the first and second columns of the identity matrix**I**._{2}Then, to interchange the first and second columns of

**A**, we postmultiply**A**by**E**, as shown below.Note that the process for performing an elementary column operation on an

*r*x*c*matrix is very similar to the process for performing an elementary row operation. The main differences are:- To operate on the
*r*x*c*matrix**A**, the row operator**E**is created from an*r*x*r*identity matrix; whereas the column operator**E**is created from an*c*x*c*identity matrix.

- To perform a row operation,
**A**is*premultiplied*by**E**; whereas to perform a column operation,**A**is*postmultiplied*by**E**.

**Problem 1**Assume that

**A**is a 4 x 3 matrix. Suppose you want to multiply each element in the second column of matrix**A**by 9. Find the elementary column operator**E**.**Solution**To find the elementary column operator

**E**, we multiply each element in the second column of the identity matrix**I**by 9._{3}

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