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### Course: Algebra (all content) > Unit 20

Lesson 3: Elementary matrix row operations# Matrix row operations

Learn how to perform the matrix elementary row operations. These operations will allow us to solve complicated linear systems with (relatively) little hassle!

## Matrix row operations

The following table summarizes the three elementary

**matrix row operations**.Matrix row operation | Example |
---|---|

Switch any two rows | |

Multiply a row by a nonzero constant | |

Add one row to another |

Matrix row operations can be used to solve systems of equations, but before we look at why, let's practice these skills.

## Switch any two rows

### Example

Perform the row operation ${R}_{1}\leftrightarrow {R}_{2}$ on the following matrix.

### Solution

So the matrix $\left[\begin{array}{rrr}{4}& {8}& {3}\\ {2}& {4}& {5}\\ 7& 1& 2\end{array}\right]$ becomes $\left[\begin{array}{rrr}{2}& {4}& {5}\\ {4}& {8}& {3}\\ 7& 1& 2\end{array}\right]$ .

Sometimes you will see the following notation used to indicate this change.

Notice how row $1$ replaces row $2$ and row $2$ replaces row $1$ . The third row is not changed.

## Multiply a row by a nonzero constant

### Example

Perform the row operation $3{R}_{2}\to {R}_{2}$ on the following matrix.

### Solution

To indicate this matrix row operation, we often see the following:

Notice here three times the second row replaces the second row. The other rows remain the same.

## Add one row to another

### Example

Perform the row operation ${R}_{1}+{R}_{2}\to {R}_{2}$ on the following matrix.

### Solution

To indicate this matrix row operation, we can write the following:

Notice how the sum of row $1$ and $2$ replaces row $2$ . The other row remains the same.

## Systems of equations and matrix row operations

Recall that in an augmented matrix, each row represents one equation in the system and each column represents a variable or the constant terms.

For example, the system on the left corresponds to the augmented matrix on the right.

System | Matrix |
---|---|

When working with augmented matrices, we can perform any of the

**matrix row operations**to create a new augmented matrix that produces an equivalent system of equations. Let's take a look at why.### Switching any two rows

Equivalent Systems | Augmented matrix |
---|---|

The two systems in the above table are equivalent, because the order of the equations doesn't matter. This means that when using an augmented matrix to solve a system, we can

**interchange any two rows**.### Multiply a row by a nonzero constant

We can multiply both sides of an equation by the same nonzero constant to obtain an equivalent equation.

In solving systems of equations, we often do this to eliminate a variable. Because the two equations are equivalent, we see that the two systems are also equivalent.

Equivalent Systems | Augmented matrix |
---|---|

This means that when using an augmented matrix to solve a system, we can

**multiply any row by a nonzero constant**.### Add one row to another

We know that we can add two equal quantities to both sides of an equation to obtain an equivalent equation.

So if $A=B$ and $C=D$ , then $A+C=B+D$ .

We do this often when solving systems of equations. For example, in this system $\begin{array}{rl}-2x-6y& =-10\\ 2x+5y& =6\end{array}$ ,
we can add the equations to obtain $-y=-4$ .

Pairing this new equation with either original equation creates an equivalent system of equations.

Equivalent Systems | Augmented matrix |
---|---|

So when using an augmented matrix to solve a system, we can

**add one row to another**.Notice that the original matrix corresponds to $\begin{array}{rl}2x+2y& =10\\ -2x-3y& =3\end{array}$ , while the final matrix corresponds to $\begin{array}{rl}x& =18\\ y& =-13\end{array}$ which simply gives the solution.

The system was solved entirely by using augmented matrices and row operations!

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