# College Algebra : Augmented Matrices

## Example Questions

### Example Question #1 : Augmented Matrices

Solve the following Augmented Matrix.

Explanation:

First step is to multiply Row 1 by , and substract Row 2, and put the resut in Row 2.

Now we have two equations.

, and

Solve for y first, and then plug the result into the other equation.

### Example Question #2 : Augmented Matrices

Write the augmented matrix for the system of equations given below and perform row operations until the augmented matrix is in row reduced echelon form, then read off the solution.

Note that interchanging rows in a matrix does not change the solution so check for interchanged rows if you don't see your augmented matrix among the augmented multiple choices.

Augmented Matrix of the System

Row Reduced Echelon Form

Solutions:

Augmented Matrix of the System

Row Reduced Echelon Form

Solutions:

Augmented Matrix of the System

Row Reduced Echelon Form

Solutions:

Augmented Matrix of the System

Row Reduced Echelon Form

Solutions:

Augmented Matrix of the System

Row Reduced Echelon Form

Solutions:

Augmented Matrix of the System

Row Reduced Echelon Form

Solutions:

Explanation:

1) Rewrite both the equations to tidy up and make sure all constant terms are isolated on one side of the equation. This will make it easier to write the augmented matrix.

2) Write the augmented matrix corresponding to the system.

Quick Refresher:

The identity matrix  has the form,

We want to perform row operations until we have reduced the augmented matrix down to the form,

where the column vector  represents what is left over from all the row operations performed to reduce the coefficient matrix down to the identity matrix. These will be the solution for  and

3) Now we can perform a succession of row operations to reduce to the desired form,

We can start by cancelling out the  in the second row by subtracting the first row from the second row to obtain:

Now we need a  to take the place of the  in the top row, so divide row 1 by :

Now we need a  to take the place of the  in the second row, so divide the second row by

Now we need a  to replace the remaining  in the first row. Subtract 2-times the second row from the first row,

4) Write the Solution Set:

### Example Question #3 : Augmented Matrices

Solve for x, y, and z.

Explanation:

From the augmented matrix, we see that

Now perform row operations to find x and y.

1. Switch Row 1 with Row 2:

2. Multiply Row 1 by-2 and add it to Row 2:

3. Multiply Row 3 by 13 and multiply Row 2 by -1. Add the two rows for a new Row 2:

4. Multiply Row 3 by -5 and add it to Row 1. Multiply Row 2 by -2 and add it to Row 1:

### Example Question #1 : Augmented Matrices

Suppose you are trying to solve a system of two linear equations in two variables using the Gauss-Jordan elimination method on an augmented matrix. After a row operation, the matrix that appears is

How many solutions does the system have?

One

Infinitely many

None

Infinitely many

Explanation:

If, during the process of Gauss-Jordan elimination, any row becomes an all zero-row, it ceases to be a factor in determining the solution set. We must examine the top row of the matrix alone.

Remember that each entry to the left of the divider represents a coefficient and each entry to the right represents a constant. Therefore, the top row in the matrix

represents the equation

,

or, equivalently,

Therefore,  can assume any real value, and . There are therefore infinitely many solutions to this equation.

### Example Question #1 : Augmented Matrices

Suppose you are trying to solve a system of two linear equations in two variables using the Gauss-Jordan elimination method on an augmented matrix. After a row operation, the matrix that appears is

How many solutions does the system have?

One

None

Infinitely many

One

Explanation:

When applying the Gauss-Jordan method to solve a two-by-two linear system, the objective is to use row operations to form an augmented matrix of the form

.

If this happens, then there is one and only one solution to the system, represented by the ordered pair . The augmented matrix given,

is in this form, and there is one solution, .

### Example Question #531 : College Algebra

Suppose you are trying to solve a system of three linear equations in three variables using the Gauss-Jordan elimination method on an augmented matrix. After a row operation, the matrix that appears is

How many solutions does the system have?

Infinitely many

One

None

None

Explanation:

The red flag in this augmented matrix is the bottom row, which has 0's as all of its entries to the left of the divider and a nonzero entry to the right. Any time this happens during the process of Gauss-Jordan elimination, this signals that the system of equations has no solution.

### Example Question #1 : Augmented Matrices

Suppose you are trying to solve a system of three linear equations in three variables using the Gauss-Jordan elimination method on an augmented matrix. After a row operation, the matrix that appears is

How many solutions does the system have?

Infinitely many

One

None

Infinitely many

Explanation:

If, during the process of Gauss-Jordan elimination, any row becomes an all zero-row, it ceases to be a factor in determining the solution set. We must examine the other rows of the matrix.

Remember that each entry to the left of the divider represents a coefficient and each entry to the right represents a constant. Therefore, the two nonzero rows in the matrix

represent the equations

and

,

respectively.

The equations can be solved for  and  in terms of :

and

Therefore,  can assume any real value, with  and  dependent on ; this is a dependent system, with infinitely many solutions.

### Example Question #1 : Augmented Matrices

When solving a system of three linear equations in three variables using the Gauss-Jordan elimination method, your initial augmented matrix is as follows:

Which of the following is notation for the next step you should perform?

Explanation:

When applying the Gauss-Jordan method to solve a three-by-three linear system, the objective is to use row operations to form an augmented matrix of the form

,

with  the solution of the system.

Once the initial augmented matrix

is set up, the first step should always be to get a 1 in the upper left position; this is usually done by multiplying every element in Row 1 by the reciprocal of the first element in that row. Since the element in that position is , every element in Row 1 is multiplied by reciprocal . This step can be written using the notation .

### Example Question #2 : Augmented Matrices

Use augmented matrices to solve the following system of equations:

Explanation:

Begin by constructing an augmented matrix for our system of equations:

Switch rows 1 and 2:

Add row 2 and twice row 1, to row 1:

Divide row 1 by 5:

Subtract 3 times row 1, from row 2:

Divide row 2 by 2:

Solution:

Plug the respective values of x and y into both equations to verify the solution:

### Example Question #3 : Augmented Matrices

Use augmented matrices to solve the following system of equations:

Explanation:

Construct an augmented matrix for our system of equations:

Swap rows 1 and 3:

Subtract 2 times row 1 from row 2:

Subtract row 1 from row 3:

Add row 2 and three times row 3, to row 3:

Divide row 3 by 10:

Add row 3 to row 1:

Subtract row 3 from row 2:

Divide row 2 by 3:

Solution:

Plug the respective values of x, y, and z into all equations to verify the solution: