Avon High School ACE COLLEGE ALGEBRA II - NOTES Mr. Record: Room ALC-129
Section:
9.1
Matrix Solutions to Linear Systems Semester 2 - Day 13
If you’ve ever worked with a program like Microsoft Excel®, then you most likely already know what a matrix is! It is simply an array of values organized into rows and columns.
A matrix can give us a shortened way of writing a system of equations. The first step in solving a system of equations using a matrix is to write an augmented matrix.
Here is an example:
3 x
 y

2 z

31 x
 y

2 z

19 x

3 y

2 z

25

3 1 2 31

1 1 2 19



 1 3 2 25


Write the solution set for a system of equations represented
by the matrix .




1
0

1
1

1
12

8
15
0 0 1 1




.
The following row operations produce matrices that represent systems with the same solution set:
1. Two rows of a matrix may be interchanged. This is the same as interchanging two equations in a
linear system.
2. The elements in any row may be multiplied by a nonzero number. This is the same as multiplying
both sides of an equation by a nonzero number.
3. The elements in any row may be multiplied by a nonzero number, and these products may be
added to the corresponding elements in any other row. This is the same as multiplying both sides
of an equation by a nonzero number and then adding equations to eliminate a variable.
Two matrices are row equivalent if one can be obtained from the other by a sequence of row operations.

Use the matrix and perform each indicated row operation:




4
1
12
6


20

3

2 1
3

8
7
9



 a. R
1

R
2 b.
1
4
R
1 c. 3 R
2

R
3
The process that we use to solve linear systems using matrix row operations is called Gaussian
elimination after German mathematician Carl Friedrich Gauss (1777-1855).
1. Write the augmented matrix for the system.
2. Use matrix row operations to simplify the matrix to a row-equivalent matrix in row-echelon form,
with 1’s down the main diagonal from upper left to lower right, and 0’s below the 1’s.

1 * * *



* * * *
* * * *
 
1 * * *




 0 * * *
  0 * * *







1 * * *
0 1 * *




 

1 * * *
0 1 * *
  0 * * *   0 0 * *
 
1 * * *




 0 1 * *
  0 0 1 *




Get 1 in the upper left-hand corner.
Use the 1 in the first column to get 0’s below it.
Get 1 in th second row, second column
Use the 1 in the second column to get
0 below it.
Get 1 in the third row, third column position. positio0n.
3. Write the system of linear equations corresponding to the matrix in Step 2 and use back
substitution to find the system’s solution.

Example 3
Use matrices to solve the system:
2 x
 y

2 z

18 x
 y

2 z

9 x

2 y
 z

6

When we use Gaussian elimination, we obtain a matrix is row-echelon form, with 1’s down the main diagonal and 0’s below the 1’s. A second method, called
Gauss-Jordan elimination , after Carl Friedrich Gauss and
Wilhelm Jordan (1842-1899), continues the process until a matrix with 1’s down the main diagonal and 0’s in every other position above and below the 1’s is found. Such a matrix is said to be in reduced row-echelon form and looks like this:

1 0 0

 0 1 0
0 0 1 a b c




Modern supercomputers are capable of solving systems with more than 600,000 variables. The augmented matrices for such systems are huge, but the process the computers follow is very much like we are doing by pencil and paper.
Begin with the Home Screen Press 7: Matrix & Vector
and choose Caclulate 1: Create --------------------------> 1:Matrix
Choose appropriate values for After entering your augmented matrix, select ctrl var to store it with a name, like abc. the number of rows and columns
Example 4
Use the TI-Nspire to solve the system:
2 w
 x

3 y z 6 w
 x

2 y

2 z
 
1 w
 x
 y
 z
 
4
 
2 x

2 y
  
7
Select Menu

7:Matrix & Vector Enter the name of your matrix
5: Reduced Row-Echelon Form after the rref(…. And press enter.