Systems of Linear Equations

### Learning Objectives

By the end of this section, you will be able to:

- Write the augmented matrix for a system of equations
- Use row operations on a matrix
- Solve systems of equations using matrices

Before you get started, take this readiness quiz.

- Solve:
If you missed this problem, review (Figure).

- Solve:
If you missed this problem, review (Figure).

- Evaluate when and
If you missed this problem, review (Figure).

### Write the Augmented Matrix for a System of Equations

Solving a system of equations can be a tedious operation where a simple mistake can wreak havoc on finding the solution. An alternative method which uses the basic procedures of elimination but with notation that is simpler is available. The method involves using a matrix. A matrix is a rectangular array of numbers arranged in rows and columns.

Matrix

A **matrix** is a rectangular array of numbers arranged in rows and columns.

A matrix with *m* rows and *n* columns has order The matrix on the left below has 2 rows and 3 columns and so it has order We say it is a 2 by 3 matrix.

Each number in the matrix is called an element or entry in the matrix.

We will use a matrix to represent a system of linear equations. We write each equation in standard form and the coefficients of the variables and the constant of each equation becomes a row in the matrix. Each column then would be the coefficients of one of the variables in the system or the constants. A vertical line replaces the equal signs. We call the resulting matrix the augmented matrix for the system of equations.

Notice the first column is made up of all the coefficients of *x*, the second column is the all the coefficients of *y*, and the third column is all the constants.

Write each system of linear equations as an augmented matrix:

ⓐⓑ

ⓐ The second equation is not in standard form. We rewrite the second equation in standard form.

We replace the second equation with its standard form. In the augmented matrix, the first equation gives us the first row and the second equation gives us the second row. The vertical line replaces the equal signs.

ⓑ All three equations are in standard form. In the augmented matrix the first equation gives us the first row, the second equation gives us the second row, and the third equation gives us the third row. The vertical line replaces the equal signs.

Write each system of linear equations as an augmented matrix:

ⓐⓑ

ⓐ

ⓑ

Write each system of linear equations as an augmented matrix:

ⓐⓑ

ⓐ

ⓑ

It is important as we solve systems of equations using matrices to be able to go back and forth between the system and the matrix. The next example asks us to take the information in the matrix and write the system of equations.

Write the system of equations that corresponds to the augmented matrix:

We remember that each row corresponds to an equation and that each entry is a coefficient of a variable or the constant. The vertical line replaces the equal sign. Since this matrix is a , we know it will translate into a system of three equations with three variables.

Write the system of equations that corresponds to the augmented matrix:

Write the system of equations that corresponds to the augmented matrix:

### Use Row Operations on a Matrix

Once a system of equations is in its augmented matrix form, we will perform operations on the rows that will lead us to the solution.

To solve by elimination, it doesn’t matter which order we place the equations in the system. Similarly, in the matrix we can interchange the rows.

When we solve by elimination, we often multiply one of the equations by a constant. Since each row represents an equation, and we can multiply each side of an equation by a constant, similarly we can multiply each entry in a row by any real number except 0.

In elimination, we often add a multiple of one row to another row. In the matrix we can replace a row with its sum with a multiple of another row.

These actions are called row operations and will help us use the matrix to solve a system of equations.

Row Operations

In a matrix, the following operations can be performed on any row and the resulting matrix will be equivalent to the original matrix.

- Interchange any two rows.
- Multiply a row by any real number except 0.
- Add a nonzero multiple of one row to another row.

Performing these operations is easy to do but all the arithmetic can result in a mistake. If we use a system to record the row operation in each step, it is much easier to go back and check our work.

We use capital letters with subscripts to represent each row. We then show the operation to the left of the new matrix. To show interchanging a row:

To multiply row 2 by :

To multiply row 2 by and add it to row 1:

Perform the indicated operations on the augmented matrix:

ⓐ Interchange rows 2 and 3.

ⓑ Multiply row 2 by 5.

ⓒ Multiply row 3 by and add to row 1.

ⓐ We interchange rows 2 and 3.

ⓑ We multiply row 2 by 5.

ⓒ We multiply row 3 by and add to row 1.

Perform the indicated operations on the augmented matrix:

ⓐ Interchange rows 1 and 3.

ⓑ Multiply row 3 by 3.

ⓒ Multiply row 3 by 2 and add to row 2.

ⓐ

ⓑ

ⓒ

Perform the indicated operations on the augmented matrix:

ⓐ Interchange rows 1 and 2,

ⓑ Multiply row 1 by 2,

ⓒ Multiply row 2 by 3 and add to row 1.

ⓐ

ⓑ

ⓒ

Now that we have practiced the row operations, we will look at an augmented matrix and figure out what operation we will use to reach a goal. This is exactly what we did when we did elimination. We decided what number to multiply a row by in order that a variable would be eliminated when we added the rows together.

Given this system, what would you do to eliminate *x*?

This next example essentially does the same thing, but to the matrix.

Perform the needed row operation that will get the first entry in row 2 to be zero in the augmented matrix:

To make the 4 a 0, we could multiply row 1 by and then add it to row 2.

Perform the needed row operation that will get the first entry in row 2 to be zero in the augmented matrix:

Perform the needed row operation that will get the first entry in row 2 to be zero in the augmented matrix:

### Solve Systems of Equations Using Matrices

To solve a system of equations using matrices, we transform the augmented matrix into a matrix in row-echelon form using row operations. For a consistent and independent system of equations, its augmented matrix is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.

Row-Echelon Form

For a consistent and independent system of equations, its augmented matrix is in **row-echelon form** when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.

Once we get the augmented matrix into row-echelon form, we can write the equivalent system of equations and read the value of at least one variable. We then substitute this value in another equation to continue to solve for the other variables. This process is illustrated in the next example.

How to Solve a System of Equations Using a Matrix

Solve the system of equations using a matrix:

Solve the system of equations using a matrix:

The solution is

Solve the system of equations using a matrix:

The solution is

The steps are summarized here.

Solve a system of equations using matrices.

- Write the augmented matrix for the system of equations.
- Using row operations get the entry in row 1, column 1 to be 1.
- Using row operations, get zeros in column 1 below the 1.
- Using row operations, get the entry in row 2, column 2 to be 1.
- Continue the process until the matrix is in row-echelon form.
- Write the corresponding system of equations.
- Use substitution to find the remaining variables.
- Write the solution as an ordered pair or triple.
- Check that the solution makes the original equations true.

Here is a visual to show the order for getting the 1’s and 0’s in the proper position for row-echelon form.

We use the same procedure when the system of equations has three equations.

Solve the system of equations using a matrix:

Write the augmented matrix for the equations. | |

Interchange row 1 and 3 to get the entry in row 1, column 1 to be 1. | |

Using row operations, get zeros in column 1 below the 1. | |

The entry in row 2, column 2 is now 1. | |

Continue the process until the matrix is in row-echelon form. | |

The matrix is now in row-echelon form. | |

Write the corresponding system of equations. | |

Use substitution to find the remaining variables. | |

Write the solution as an ordered pair or triple. | |

Check that the solution makes the original equations true. | We leave the check for you. |

Solve the system of equations using a matrix:

Solve the system of equations using a matrix:

So far our work with matrices has only been with systems that are consistent and independent, which means they have exactly one solution. Let’s now look at what happens when we use a matrix for a dependent or inconsistent system.

Solve the system of equations using a matrix:

Write the augmented matrix for the equations. | |

The entry in row 1, column 1 is 1. | |

Using row operations, get zeros in column 1 below the 1. | |

Continue the process until the matrix is in row-echelon form. | |

Multiply row 2 by 2 and add it to row 3. | |

At this point, we have all zeros on the left of row 3. | |

Write the corresponding system of equations. | |

Since we have a false statement. Just as when we solved a system using other methods, this tells us we have an inconsistent system. There is no solution. |

Solve the system of equations using a matrix:

no solution

Solve the system of equations using a matrix:

no solution

The last system was inconsistent and so had no solutions. The next example is dependent and has infinitely many solutions.

Solve the system of equations using a matrix:

Write the augmented matrix for the equations. | |

The entry in row 1, column 1 is 1. | |

Using row operations, get zeros in column 1 below the 1. | |

Continue the process until the matrix is in row-echelon form. | |

Multiply row 2 by and add it to row 3. | |

At this point, we have all zeros in the bottom row. | |

Write the corresponding system of equations. | |

Since we have a true statement. Just as when we solved by substitution, this tells us we have a dependent system. There are infinitely many solutions. | |

Solve for y in terms of z in the second equation. | |

Solve the first equation for x in terms of z. | |

Substitute | |

Simplify. | |

Simplify. | |

Simplify. | |

The system has infinitely many solutions where is any real number. |

Solve the system of equations using a matrix:

infinitely many solutions where is any real number.

Solve the system of equations using a matrix:

infinitely many solutions where is any real number.

Access this online resource for additional instruction and practice with Gaussian Elimination.

### Key Concepts

**Matrix:**A matrix is a rectangular array of numbers arranged in rows and columns. A matrix with*m*rows and*n*columns has*order*The matrix on the left below has 2 rows and 3 columns and so it has order We say it is a 2 by 3 matrix.Each number in the matrix is called an

*element*or*entry*in the matrix.**Row Operations:**In a matrix, the following operations can be performed on any row and the resulting matrix will be equivalent to the original matrix.- Interchange any two rows
- Multiply a row by any real number except 0
- Add a nonzero multiple of one row to another row

**Row-Echelon Form:**For a consistent and independent system of equations, its augmented matrix is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.(Video) Intermediate Algebra: Part 7 (Systems of Equations with Matrices)**How to solve a system of equations using matrices.**- Write the augmented matrix for the system of equations.
- Using row operations get the entry in row 1, column 1 to be 1.
- Using row operations, get zeros in column 1 below the 1.
- Using row operations, get the entry in row 2, column 2 to be 1.
- Continue the process until the matrix is in row-echelon form.
- Write the corresponding system of equations.
- Use substitution to find the remaining variables.
- Write the solution as an ordered pair or triple.
- Check that the solution makes the original equations true.

#### Practice Makes Perfect

**Write the Augmented Matrix for a System of Equations**

In the following exercises, write each system of linear equations as an augmented matrix.

ⓐ

ⓑ

ⓐ

ⓑ

ⓐ

ⓑ

ⓐ

ⓑ

ⓐ

ⓑ

ⓐ

ⓑ

Write the system of equations that corresponds to the augmented matrix.

**Use Row Operations on a Matrix**

In the following exercises, perform the indicated operations on the augmented matrices.

ⓐ Interchange rows 1 and 2

ⓑ Multiply row 2 by 3

ⓒ Multiply row 2 by and add row 1 to it.

ⓐ Interchange rows 1 and 2

ⓑ Multiply row 1 by 4

ⓒ Multiply row 2 by 3 and add row 1 to it.

ⓐ

ⓑ

ⓒ

ⓐ Interchange rows 2 and 3

ⓑ Multiply row 1 by 4

ⓒ Multiply row 2 by and add to row 3.

ⓐ Interchange rows 2 and 3

ⓑ Multiply row 2 by 5

ⓒ Multiply row 3 by and add to row 1.

ⓐ

ⓑ

ⓒ

Perform the needed row operations that will get the first entry in both row 2 and row 3 to be zero in the augmented matrix:

**Solve Systems of Equations Using Matrices**

In the following exercises, solve each system of equations using a matrix.

In the following exercises, solve each system of equations using a matrix.

no solution

no solution

infinitely many solutions where is any real number

infinitely many solutions where is any real number

#### Writing Exercises

Solve the system of equations ⓐ by graphing and ⓑ by substitution. ⓒ Which method do you prefer? Why?

Solve the system of equations by substitution and explain all your steps in words.

Answers will vary.

#### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

### Glossary

- matrix
- A matrix is a rectangular array of numbers arranged in rows and columns.

- row-echelon form
- A matrix is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.

## FAQs

### How do you solve a system of equations using matrices? ›

A system of equations can be solved using **matrix multiplication**. A is the coefficient matrix, X the variable matrix and B the constant matrix. The second method to find the solution for the system of equations is Row reduction or Gaussian Elimination. The augmented matrix for the linear equations is written.

### How do you solve a system of equations using matrices on a TI 84? ›

Into the top row. So I'll put my coefficients. And my equation my equal on the top row and then do

### How do you solve a 2 by 2 system using matrices? ›

And once we have our matrix equation. We want to solve this equation for matrix x to do this we're

### How do you solve 3 equations with 3 variables using matrices? ›

Terms the variable matrix x will contain the variables used in the system. And matrix b will be the

### How do you solve a system of equations with 4 variables using matrices? ›

Gaussian Elimination With 4 Variables Using Elementary Row ... - YouTube

### How do you solve a 3x4 matrix on a TI 84? ›

Using the TI-84 RREF feature to solve a 3x4 System with a Single Solution

### How do you solve matrices with variables? ›

Solving Variables in Equal Matrices (Equivalent Matrices) [fbt] - YouTube

### How do you solve a matrix on a TI 83 Plus? ›

Putting Matrices into Ti-83 - YouTube

### How do you solve a matrix step by step? ›

Solving Matrix Equations - YouTube

### How do you solve a 3 by 4 matrix? ›

Solving Matrices by Hand 3x4 - YouTube

### How do you solve 3 variable equations quickly? ›

Solving a System of Equations Involving 3 Variables ... - YouTube

### How do you solve a matrix using the inverse method? ›

**SOLVING A SYSTEM OF EQUATIONS USING THE INVERSE OF A MATRIX**

- Given a system of equations, write the coefficient matrix A, the variable matrix X, and the constant matrix B. Then.
- AX=B.
- Multiply both sides by the inverse of A to obtain the solution.

### How do you solve a 4 by 4 matrix? ›

How To Find The Determinant of a 4x4 Matrix - YouTube

### How do you solve 4 equations with 4 unknowns on a calculator? ›

Solve 4 by 4 system of linear equations by Canon F-789SGA ...

### Is it possible to solve 3 equations with 4 variables? ›

Solving a system of 3 equations and 4 variables using matrix row-echelon form. **Sal solves a linear system with 3 equations and 4 variables by representing it with an augmented matrix and bringing the matrix to reduced row-echelon form**.

### What are the 3 methods for solving a system of equations? ›

There are three ways to solve systems of linear equations in two variables: **graphing**. **substitution method**. **elimination method**.

### What is the easiest way to solve system of equations? ›

There are three methods used to solve systems of equations: **graphing, substitution, and elimination**. To solve a system by graphing, you simply graph the given equations and find the point(s) where they all intersect. The coordinate of this point will give you the values of the variables that you are solving for.

### How do you find the inverse of a 3x3 matrix on a calculator? ›

Determining Inverse Matrices on the TI83/84 - YouTube

### How do you do Gauss Jordan elimination on a calculator? ›

TI 84 Gauss Jordan - YouTube

### How do you find the inverse of a 3x3 matrix on a scientific calculator? ›

Casio fx-991es: How to Find The Inverse of a Matrix - YouTube

### How do you solve a matrix in linear algebra? ›

Matrices to solve a system of equations | Khan Academy - YouTube

### What is matrix formula? ›

A matrix equation is an equation of the form **Ax = b** , where A is an m × n matrix, b is a vector in R m , and x is a vector whose coefficients x 1 , x 2 ,..., x n are unknown.

### How do you solve a matrix by hand? ›

How to Solve Equations with a Matrix - YouTube

### How do you enter a matrix into a calculator? ›

TI-84 Plus Graphing Calculator Guide: Matrices - YouTube

### Can you do matrices on TI-84? ›

**You can enter and store matrices on your TI-84 Plus calculator**. A matrix is a rectangular array of numbers arranged in rows and columns. The dimensions, r x c, of a matrix are defined by the number of rows and columns in the matrix.

### How do you clear matrices on a TI-84? ›

**Scroll to “Mem Mgmt/Del.”** **Press the “ENTER” key.** **Press “5” to select "Matrix" and press the “ENTER” key.** **Scroll to each matrix and press “DEL.”** This will clear the matrix out of the memory.

### How do you use the matrix method? ›

Matrix Method for Solving Systems of Equations - YouTube

### What is matrix with example? ›

Type of Matrix | Representation Details | Example |
---|---|---|

Hermitian Matrix | A = A θ . | Q = [ a b + i c b − i c d ] |

Skew – Hermitian Matrix | A = − A θ . | Q = [ 0 − 2 + i 2 − i 0 ] |

Orthogonal Matrix | A × A T = I | – |

Idempotent Matrix | A 2 = A | – |

### How do you solve a matrix with 2 equations? ›

Ex 2: Solve a System of Two Equations Using a Matrix Equation

### How do you square a 3x3 matrix? ›

Square 3x3 matrices faster - YouTube

### How do I find the determinant of a 3x3 matrix? ›

Determinant of 3x3 matrix - YouTube

### What is the shortcut to find the determinant of a 4x4 matrix? ›

How to Find The Determinant of a 4x4 Matrix (Shortcut Method) - YouTube

### How many equations do you need to solve for 3 variables? ›

Equations with two variables require two equations to have a unique solution (one ordered pair). So it should not be a surprise that equations with three variables require a system of **three equations** to have a unique solution (one ordered triplet).

### How do you solve a system of equations with 3 variables using substitution? ›

How to Solve Systems by Substitution With 3 Variables : Algebra - YouTube

### How do you solve a multivariable equation? ›

multivariable equations (KristaKingMath) - YouTube

### How do you solve a 2x1 matrix? ›

Multiplying Matrices 2x2 by 2x1 - Corbettmaths - YouTube

### How do you find the unknown matrix? ›

In a matrix equation, the unknown is a matrix. This means that you will denote the unknown matrix as matrix X. To solve, **check that the matrix is invertible, if it is, premultiply (multiply to the left) both sides by the matrix inverse of A**.

### What is the inverse of a matrix used for? ›

The inverse matrix is used to solve the system of linear equations. It also tells us the consistent or inconsistent behaviour of the solution of equations.

### How do you solve a system of linear equations using determinants? ›

**Solve a system of two equations using Cramer's rule.**

- Evaluate the determinant D, using the coefficients of the variables.
- Evaluate the determinant D x . D x . ...
- Evaluate the determinant D y . D y . ...
- Find x and y. ...
- Write the solution as an ordered pair.
- Check that the ordered pair is a solution to both original equations.

### How do you solve matrices with variables? ›

Solving Variables in Equal Matrices (Equivalent Matrices) [fbt] - YouTube

### How do you solve a system of three linear equations with determinants? ›

Algebra - Solving Linear Equations using Determinants 3/3 - YouTube

### Is determinant method and Cramer's rule same? ›

Cramer's rule is one of the important methods applied to solve a system of equations. In this method, the values of the variables in the system are to be calculated using the determinants of matrices. Thus, **Cramer's rule is also known as the determinant method**.

### How do you tell if a system of equations has no solution using determinant? ›

**If the determinant of a matrix is zero**, then the linear system of equations it represents has no solution.