Avon High School
Section: 9.1
ACE COLLEGE ALGEBRA II - NOTES
Matrix Solutions to Linear Systems
Mr. Record: Room ALC-129
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.
Augmented Matrices
A matrix can give us a shortened way of writing a system of equations.
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
Example 1
The first step in solving a system of
3 1 2 31
1 1 2 19 


1 3 2 25
Solving a System Using a Matrix
Write the solution set for a system of equations represented
by the matrix .
8 
1 1 1
0 1 12 15 .


0 0
1
1 
Matrix Row Operations
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.
Example 2
Performing Matrix Row Operations
Use the matrix and perform each indicated row operation:
 4 12 20 8 
 1 6 3 7 


 3 2 1 9 
1
a. R1  R2
b. R1
4
c. 3R2  R3
Gaussian Elimination
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).
Solving Linear Systems Using Gaussian Elimination
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 * * * 1 * * * 1 * * * 1 * * *
* * * *  0 * * *  0 1 * *  0 1 * *  0 1 * *

 
 
 
 

* * * * 0 * * * 0 * * * 0 0 * * 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
positio0n.
Use the 1 in
the second
column to get
0 below it.
Get 1 in the third
row, third
column position.
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
Gaussian Elimination with Back-Substitution
Use matrices to solve the system:
2x 
y  2 z  18
x  y  2z  9
x  2y  z  6
Gauss-Jordan Elimination
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 a 
0 1 0 b 


0 0 1 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.
Using TI-Nspire to Find Reduced Row-Echelon Form
Begin with the Home Screen Press 7: Matrix & Vector
and choose Caclulate
1: Create --------------------------> 1:Matrix
Choose appropriate values for
the number of rows and columns
After entering your augmented matrix, select ctrl var to store it with a name, like abc.
Example 4
Use the TI-Nspire to solve the system:
2w  x  3 y  z  6
w  x  2 y  2 z  1
w  x  y  z  4
 w  2 x  2 y  z  7
Select Menu  7:Matrix & Vector
5: Reduced Row-Echelon Form
Enter the name of your matrix
after the rref(…. And press enter.