PRE-CALCULUS NOTES
SECTION 7.4 MATRICES AND SYSTEMS OF EQUATIONS
A.) Matrix – used for solving systems of linear equations
1.) Systems of Equations – 2 or more equations in 2 or more variables.
4x  y  3
x  2y 1
x  3y  z  1
2x  y  2z  2
Examples:
x  2 y  3 z  1
rectangular array of real #’s
If m and n are positive integers, an m n matrix is a rectangular array:
a1n 
 a11

 m rows


a

a
mn 
 m1
n columns
in which each entry is a real number.
Definitions:
1.) Square Matrix – occurs when m = n.
2.) Row Matrix – matrix with only 1 row.
3.) Column Matrix – matrix with only 1 column.
How do we setup a matrix for a system of linear equations?
5
x  4 y  3z  5
 1 4 3 
 1 4 3




 x  3 y  z  3
3 
 1 3 1
 1 3 1 
 2 0 4
 2 0 4 
2x
 4z  6
6 



Coefficient Matrix
Augmented Matrix
When working with systems in Algebra we can use elementary row operations to solve
matrices:
B.) Elementary Row Operations: Under Matrix and Math
1.) Interchange 2 rows.
2.) Multiply a row by a nonzero constant.
3.) Add a multiple of a row to another row.
An elementary row operation on an augmented matrix of a given system of linear
equations produces a new augmented matrix corresponding to an equivalent system of
linear equations.
4.) Row Equivalent Matrix – if 1 matrix can be obtained from the other by a sequence of
elementary row operations.
row  (
row(
row  (
rowSwap (
ref (
rref (
Row operations help us to eliminate variables in a systems in more than 2 variables so we
can use back substitution.
C.) Row-Echelon Form (ref) – “hair-step” pattern with leading coefficients of 1.
1.) All rows consisting entirely of zeros occur at the bottom of the matrix.
2.) For each row that does not consist entirely of zeros, the 1st nonzero entry is 1
(called a leading 1 )
3.) For 2 successive (nonzero) rows, the leading 1 in the higher row is farther to the left
than the leading 1 in the lower row.
D.) Reduced-Row-Echelon Form (rref) – every column that has a leading 1 has zeros in
every position above and below its leading 1.
E.) The process of converting a matrix to an equivalent matrix in ref form by using
elementary row operations is called GAUSSIAN ELIMINATION.
To solve systems use BACK SUBSTITUTION:
Example:
x  2 y  3z  9
 x  3 y  4
2 x  5 y  5 z  17
( use ref form)
 1 2 3

 1 3 0
 2 5 5

9

4 
17 
GAUSSIAN ELIMINATION WITH BACK SUBSTITUTION:
1.) Write augmented matrix of system of linear equations (enter into calc).
2.) Rewrite matrix using elementary row operations into ref form (ref on calc).
3.) Write system of linear equations corresponding to ref matrix and use back
substitution to find the solution.
Another method employs rref form – more direct approach.
F.) Gauss-Jordan Elimination – using elementary row operations to convert a matrix
into rref form which directly gives the solution.
Example:
9
x  2 y  3z  9
 1 2 3


 x  3 y  4
4 
 1 3 0
 2 5 5
2 x  5 y  5 z  17
17 

Recall - Systems of Linear Equations have 3 possibilities:
1.) There is exactly 1 solution.
2.) There are infinitely many solutions.
3.) There is NO solution.
A system with no solutions:
After reducing we get a row of the matrix with all zeros except for the last entry. (not
possible , inconsistent system)
A system with infinitely many solutions:
If one of the columns to the left of the augmentation bar has no nonzero (pivot) element,
then we have an infinite # of solutions.
Otherwise, we have a unique solution.