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PreCalculus
6-1 Multivariable Linear Systems and Row Operations
Name:___________________
We can convert a system of linear equations into an equivalent triangular or row-echelon form (ref).
We do this using Gaussian Elimination. (See key concept on page 364)
* Operations that Produce Equivalent Systems:
Each of the following operations produces an equivalent system of linear equations.
 Interchange any two equations
 Multiply one of the equations by a nonzero real number
 Add a multiple of one equation to another equation
Ex. 1: Write the system of equations in triangular form using Gaussian Elimination. Then solve.
x  3y  2z  5
a.
3x  y – 2z  7
2x  2y  3z  3
x  2 y  3 z  28
b.
3x  y  2 z  3
 x  y  z  5
PreCalculus Ch 6A Notes_Page 1
* The augmented matrix is the matrix containing the coefficients and constant terms of a system.
Ex. 2: Write the augmented matrix for the system of linear equations.
x yz 5
a. 2 w  3 x  z  2
2w  x  y  6
4 w  5 x  7 z  11
b.  w  8 x  3 y  6
15 x  2 y  10 z  9
*Elementary Row operations (see key concept on page 366): Each of the following row operations
produces an equivalent augmented matrix.
 Interchange any 2 rows
 Multiply one row by a nonzero real number
 Add a multiple of one row to another row
*See page 367 for comparison of Gaussian Elimination and Augmented matrix.
* Row-Echelon Form (ref): A matrix is in row-echelon form if the following conditions are met.
 Rows consisting entirely of zeros (if any) appear at the bottom of the matrix.
 The first nonzero entry in a row is 1, called a leading 1.
 For 2 successive rows with nonzero entries, the leading 1 in the higher row is
farther to the left than the leading 1 in the lower row.
Ex. 3: Determine whether each matrix is in row-echelon form.
1 0 0 4 
1 0 12 
1  3 4 


a. 
b.  0 0 0 0 
c.  0 1  9 

0 1 2
0 0 1  2
0 0 0 




 1 0 5 12 


d.  0 0 1  9 
0 1 0 2 


To solve a system using an augmented matrix or Gaussian elimination, use row operations to transform
the matrix so that it’s in row-echelon form. Then write the corresponding system and use substitution to
finish solving the system. Remember, if you encounter a false equation, the system has no solution.
Ex. 4: Three families ordered meals of hamburgers (HB), French fries (FF) and drinks (DR). The items
they ordered and their total bills are shown below. Write and solve a system of equations to determine the
cost of each item.
Family
HB FF DR Total ($)
Tau
6
5
5
55
Jenkins
2
1
1
15
Hernandez 1
3
2
18
PreCalculus Ch 6A Notes_Page 2
You Try: Solve the system using augmented matrix and row-echelon form.
3 x  2 y  z  22
4 x  z  31
2 x  5 y  24
Gauss-Jordan Elimination: The process of transforming an augmented matrix to Reduced RowEchelon Form (rref), a matrix in which the first nonzero element of each row is 1 and the rest of the
elements in the same column as this element are zero.
Ex. 5: Solve the system of equations.
x yz 3
x  2 y  z  2
2 x  3 y  3z  8
When solving a system of equations, if a matrix cannot be written in reduced row-echelon form (rref),
then the system has no solution or infinitely many solutions.
PreCalculus Ch 6A Notes_Page 3
Ex. 6: Solve each system of equations.
2x  3y  z  1
a.
x  y  2z  5
x  2y  z  8
x  2y  z  8
b.
2 x  3 y  z  13
x  y  2z  5
When a system has fewer equations than variables, the system has either no solution or infinitely many
solutions. When checking your solutions, be sure to check them into the original equations.
PreCalculus Ch 6A Notes_Page 4
Ex. 7: Solve the system of equations.
4w  x  2 y  3z  10
a. 3w  4 x  2 y  8 z  3
w  3 x  4 y  11z  11
3w  x  2 y  3z  14
b.  w  x  10 y  z  11
2w  x  4 y  2 z  9
PreCalculus Ch 6A Notes_Page 5
PreCalculus
6-2 Matrix Multiplication, Inverses and Determinants
Name:___________________
*Matrix Addition is simply adding the elements in the same positions. Scalar Multiplication is
multiplying every element in a matrix by the scalar number.
*To multiply matrices A and B, the number of columns in A must equal the number of rows in B. The
product matrix has dimensions of the number of rows in A by the number of columns in B. Look at the
position of the elements to multiply. For each element in the product matrix, take the sum of the product
of the elements. For row 1, column 1, multiply all of the elements in row 1 by all of the elements in
column 1 and add them.
* The product of two matrices A and B is defined when the number of __________ in matrix A is the
same to the number of _____________ in matrix B.
 A   B   AB
( m  n)  ( n  p )  ( m  p )
 4 5
 7 0 1

Ex. 1: Use matrices A   1
9  and B  
 to find each product, if possible.
 2 6 8
 6 3 


a. AB
b. BA
Properties of Matrix Multiplication: For any matrices A, B, and C for which the matrix product is
defined and any scalar k, the following properties are true.
 Associative Property of Matrix Multiplication
(AB)C=A(BC)
 Associative Property of Scalar Multiplication
k(AB)=(kA)B = A(kB)
 Left Distributive Property
C(A +B) = CA + CB
 Right Distributive Property
(A +B)C = AC + BC
*No Commutative Property on matrix multiplication : AB  BA
Ex. 2: The number of touchdowns (TD), field goals (FG), and point after touchdown (PAT), and twopoint conversions (2EP) for the three top teams in the high school league for this season is shown below.
The other table shows the number of points each type of score is worth. Use the information to determine
the team that scored the most points.
Team
Tigers
Rams
Eagles
TD
27
24
21
FG
7
12
14
PAT
21
18
12
PreCalculus Ch 6A Notes_Page 6
2EP
2
3
9
Score
TD
FG
PAT
2EP
Points
6
3
1
2
*The Identity Matrix is an (n x n) square matrix and has 1’s along the main diagonal and 0’s
everywhere else.
Ex. 3: Write the system of equations as a matrix equation, AX=B. Then use Gauss-Jordan Elimination on
the augmented matrix to solve for X.
2 x1  2 x2  3x3  3
a.
x1  3x2  2 x3  5
3x1  x2  x3  4
x1  x2  x3  2
b.
2 x1  x2  2 x3  4
 x1  4 x2  x3  3
PreCalculus Ch 6A Notes_Page 7
*Inverses of a Square matrix: Let A be an (n x n) matrix.
If there exists a matrix B such that AB = BA = In , then B is called the inverse of A and is written A-1.
So AA1  A1 A  I n .
Ex. 4: Determine whether A and B are inverse matrices.
 5 6
 5 6 
a. A  
 and B  

 4 5
 4 5 
 6 2
 1 2 
b. A  
 and B  

2 1
 2 6 
If a matrix A has an inverse, it is invertible, or nonsingular.
If a matrix A does not have an inverse, it is singular.
*One method for finding the inverse is to use a doubly augmented matrix. (See page 379)
Ex. 5: Find A-1, if it exists. If A-1 does not exist, write singular.
 4 6 
a. A  

 3 5 
 12 9 
b. A  

 4 3 
PreCalculus Ch 6A Notes_Page 8
* Second method for finding the inverse of a 2 x 2 matrix:
a b 
Let A  
: A is invertible if and only if ad – bc ≠ 0.

c d 
1  d b 
If A is invertible, then A1 


ad  bc  c a 
The number (ad – bc) is called the determinant of the 2 x 2 matrix
a b
and is denoted det  A  A 
 ad  bc .
c d
Ex. 6: Find the determinant of each matrix. Then find the inverse of the matrix, if it exists.
 5 10 
a. A  

 4 8 
 2 4 
b. B  

 4 6 
*How to find the determinant of a 3 x 3 matrix:
a b c 
e f
d f
d e


Let A   d e f  then det  A  A  a
b
c
h i
g i
g h
g h i 


*Recall: The diagonal method for finding determinant of a 3×3 Matrix:
a b c  a b c


det  d e f   d e f
 (aei  bfg  cdh)  ( gec  hfa  idb)
g h i  g h i


*We will use the graphing calculator to find the inverse of a 3 x 3 matrix, if it exists.
Ex. 7: Find the determinant of matrix. Then find the inverse, if it exists.
a.
 3 1 0


D   2 1 4 
 1 2 5


PreCalculus Ch 6A Notes_Page 9
b.
  1 2 1 


C  4 0
3
  3 1 2 


PreCalculus Ch 6A Notes_Page 10