Download College Algebra Lecture Notes, Section 6.1

College Algebra Lecture Notes
Section 6.1
Page 1 of 7
Section 6.1: Solving Systems Using Matrices and Row Operations
Big Idea: A system of three equations in three unknowns can be solved more efficiently using
elimination by only writing the coefficients of the equations when doing row operations
Big Skill: You should be able to convert a system of equations into an augmented matrix, and
then solve the system by performing row operations on the matrix.
A matrix is a set of numbers arranged into rows and columns to form a rectangular array.
An augmented matrix is a matrix formed by copying the coefficients of a system of equations
into a square matrix and then copying the constants for the system into a column next to that
square matrix. It is common to put a vertical dashed or solid line separating the square matrix
and the column of constants.
Example of a system of three equations in three unknowns (that happens to be in
triangular form) and that system’s augmented matrix:
 1 2 1 1 
x  2 y  z  1



y  2 z  5  0 1 2 5


 0 0 1 3
z  3

Note that the position of each number in the matrix relates to its meaning in the system of
equations.
 Every number in the first column is an x coefficient.
 Every number in the second column is a y coefficient.
 Every number in the third column is a z coefficient.
 The vertical line represents all the equals signs.
 Every number in the final column is a constant.
 All the numbers in the first row correspond to the coefficients and constant in the first
equation
 Etc. for the other rows.
The goal of this section is to convert a linear system of equations into an augmented matrix, and
then perform row operations on the matrix to get the matrix into triangular form. This is
equivalent to getting the system of equations into triangular form because the position of each
number in the matrix corresponds to a coefficient or constant in the equations. We use matrices
to do this because it is more efficient when we don’t have to write all the x’s, y’s, z’s, and =’s,
and because it will lead us to more advanced matrix techniques later.
College Algebra Lecture Notes
Section 6.1
Page 2 of 7
Elementary Row Operations on a Matrix

Any two rows can be interchanged (without changing the solution of the underlying
system of equations).

Every element of any row can be multiplied by a nonzero constant (without changing the
solution of the underlying system of equations).

The sum of any two rows can be used to replace one of the two rows added together
(without changing the solution of the underlying system of equations).
Steps for Solving a System of Three Linear Equations in Three Unknowns Using an
Augmented Matrix

Eliminate the x coefficients from the second and third rows using elimination.

Eliminate the y coefficient from the third row using elimination.

Make the coefficient of z in the third equation equal to one by dividing by the appropriate
number.

Convert the triangular matrix back into a system of equations.

Substitute z into the second equation to find the solution for y, then substitute y and z into
the first equation to find the solution for x.
Rules for showing your work:

Draw an arrow from one transformed matrix to the next, and write on the arrow what you
did to transform the matrix.

Any row that is unchanged gets copied from one system to the next.
College Algebra Lecture Notes
Section 6.1
Page 3 of 7
Example:
y  z  2
 x 

 x  2 y  3 z  12
2 x  2 y  z  9

 convert to augmented matrix
2 2 2

 0 1 4
 0 4 1
 (4) R 2
1 1 1

1 2 3
0 2 1
 (1) R1
2 2 2
4


 0 4 16 56 
 0 4 1 5
 R 2  R3  R3
2 

12 
9 
4

14 
5
1 1 1 2 


 1 2 3 12 
 2 2 1 9 
 R1  R 2  R 2
2 2 2

 0 4 16
 0 0 17
 R3  (17)
1 1 1

 0 1 4
 2 2 1
2 2 2
4


 0 4 16 56 
 0 0
1
3
 convert back to a system of equations
2

14 
9 
 (2) R1
2 2 2 4 


 0 1 4 14 
 2 2 1 9 
 R1  R3  R3
2 2 2

 0 1 4
 2 4 1


4

14 
5
4

56 
51
2 x  2 y  2 z  4

4 y  16 z  56


z  3

Substitute into equation #2:
4 y  16  3  56  y  2
Substitute into equation #1:
2 x  2  2   2  3  4  x  1
Using row operations to get a matrix into triangular form is called Gaussian elimination.
A matrix in triangular form is also said to be in row-ecehelon form.
College Algebra Lecture Notes



Page 4 of 7
Using continued row operations to get a matrix with 1’s on the diagonal and zeros
everywhere else except for the constants column is called Gauss-Jordan elimination.
This corresponds to back-substitution, which solves for each variable separately.
A matrix in this form is said to be in reduced row-echelon form.
Example:
2 2 2
4


 0 4 16 56 
 0 0
1
3
 (16) R3  R 2  R3
2 2 2

0 4 0
 0 0 1
 R2  4
4

8
3
2 2 2 4 


2
0 1 0
 0 0 1 3
 2 R 2  2 R3+R1  R1
2 0 0

0 1 0
 0 0 1
2

2
3
 R1  (2)
1 0 0 1


0 1 0 2 
0 0 1 3
 convert back to a system of equations
x




Section 6.1
 1
y

2
z  3
College Algebra Lecture Notes
Practice:
 2x  y  2
1. 
4 x  3 y  9
y  z  0
 x 

2.  x  2 y  5 z  3
 3x 
y  z  6

Section 6.1
Page 5 of 7
College Algebra Lecture Notes
Section 6.1
 x  3y  z  8

3.  3x  y  6 z  12
4 x  y  2 z  1

 x  2 y  5 z  4

4.  x
 2z  0
 4 x  2 y  11z  2

Page 6 of 7
College Algebra Lecture Notes
z  3
 x  2y 

5. 2 x  5 y  6 z  7
2 x  3 y  2 z  5

Section 6.1
Page 7 of 7