Download Precalculus_Unit 5 extension_2016_2017

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AMHS Precalculus - Unit 5 extension
Matrices
A matrix is a rectangular array of variables or constants in horizontal rows and vertical columns, usually
enclosed in brackets. In a matrix, the numbers or data are organized so that each position in the matrix
has a purpose. Each value in the matrix is called an element. A matrix is usually named using an
uppercase letter.
1 4 3 
A

0 2 9 
A matrix can be described by its dimensions. A matrix with m rows and n columns is an m x n matrix
(read “m by n”). Matrix A is a 2 X 3 matrix because it has 2 rows and 3 columns.
a12 refers to an element of A that is in row 1 column 2, a12  4 .
Operations with Matrices
Matrices can be added or subtracted if they have the same dimensions. The corresponding elements are
added or subtracted.
Addition
a b   e
c d    g

 
Subtraction
f  a  e b  f 

h  c  g d  h 
a b   e
c d    g

 
f  a  e b  f 

h  c  g d  h 
You can multiply any matrix by a constant called a scalar. When you do this, you multiply each individual
element by the value of the scalar. This operation is called scalar multiplication.
Scalar Multiplication
 a b   ka kb 
k
   kc kd 
c
d

 

For any matrices A, B, and C for which the matrix sum and product are defined and any scalar k, the
following properties are true.



Commutative Property of Addition
Associative Property of Addition
Scalar Distributive Property
A+B=B+A
(A + B) + C = A + (B + C)
k(A + B) = kA + kB
or
(A + B)k =kA + kB
And multi-step operations can be performed on matrices. The order of operations is the same as with
real numbers.
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AMHS Precalculus - Unit 5 extension
Ex 1: The drink menu from a fast-food restaurant is shown below. The store owner has decided that all
of the prices must be increased by 10%.
Drink
Soda
Iced Tea
Lemonade
Coffee
Small
$0.95
$0.75
$0.75
$1.00
Medium
$1.00
$0.80
$0.80
$1.10
Large
$1.05
$0.85
$0.85
$1.20
a. Write matrix C to represent the current prices. State the dimensions of matrix C.
b. Use scalar multiplication to determine matrix N to represent the new prices. State the
dimensions of matrix N.
c. What is P = N – C ? What does P represent in this situation?
d. What is p32 and what does it represent?
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AMHS Precalculus - Unit 5 extension
Matrix Multiplication
You can multiply two matrices A and B if and only if the number of columns in A is equal to the number
of rows in B.
The product of two matrices is found by multiplying columns and rows,
summing products as follows:
A

B
=
AB
f   ae  bg

h   ce  dg
a b   e
c d    g

 
af  bh 
cf  dh 
Using the below matrices,
1 3 5
A

 4 2 0 
 2 3 
B   2 8 


 1 7 
Ex 1:
BA and AB
Ex 2:
CD and DC
Ex 3:
AC and CA
 4 1
C

3 0 
 3 6 
D

 4 5 
*Is matrix multiplication commutative? Justify your answer.
Find each product…
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AMHS Precalculus - Unit 5 extension
Inverse Matrices and Systems of Equations

Identity Matrix is a square matrix that, when multiplied by another matrix, equals that same
matrix.
2 X 2 Identity Matrix
3 X 3 Identity Matrix
1 0 0 
I  0 1 0 


0 0 1 
1 0
I 

0 1 
Identity Matrix for Multiplication
 a b  1 0  1 0   a b   a b 
 c d   0 1   0 1  c d    c d 

 
 

 

Two n x n matrices are inverses of each other if their product is the identity matrix.
If matrix A has an inverse symbolized A1 , then A  A1  A1  A  I
Ex 1:
a.
b.
Determine whether the matrices are inverses.
 4 2
A

 2 1 
5
3
C

 2 6 
1

B  4
1
 2
3

D  4
1
 4

1
 
2
1 

5
8

3
8 
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AMHS Precalculus - Unit 5 extension

a b 
 is
c d 
1  d b 
where the determinant ad  bc  0.
A1 
ad  bc  c a 
The inverse of a matrix A  
If the determinant of a matrix is 0, the matrix does not have an inverse.
*Note: If the matrix is not square, the determinant and inverse cannot be found.
7 5

 2 1
Ex 1:
Find the inverse of matrix P, if it exists. P  
Ex 2:
 2 3 1

Using your graphing calculator, find the inverse of matrix A, if it exists. A  0
2 4 

 2 5 6 

Matrix Equations:
Matrices can be used to represent and solve systems of equations. Consider the system of
equations below. You can write a matrix equation to solve the system.
x  2y  9
3x  6 y  3
 x  2 y  9 
3 x  6 y    3 

  
Write the left side of the matrix equation as the product of the coefficient matrix and the
variable matrix. Write the right side as a constant matrix.
A
 X = B
1 2   x  9
3  6    y    3 

    
Then solve the matrix equation using A1 .
AX  B
A1 AX  A1 B
IX  A1 B
X  A1 B
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AMHS Precalculus - Unit 5 extension
10 x  7 y  59
6 x  5 y  63
Ex 1:
Solve the system using an inverse matrix.
Ex 2:
4 x  5 y  6 z  14
Solve the system using an inverse matrix and a graphing calculator. 3 x  2 y  7 z  47
7 x  6 y  8 z  15
Ex 3:
The sum of three numbers is 14. The largest is 4 times the smallest, while the sum of the
smallest and twice the largest is 18. Find the numbers.
Augmented Matrices and Systems of Equations
An augmented matrix contains the coefficient matrix with an extra column containing the constant
terms. You can use a graphing calculator to reduce the augmented matrix so that the solution of the
system of equations can be easily determined.
Step 1: Write the augmented matrix from the system of equations and enter it into the calculator.
Step 2: Find the reduced row echelon form (rref) using the graphing calculator.
*View the reduced row echelon form matrix as an augmented matrix, where the identity matrix is now
the coefficient matrix and the last column of constants are the solutions to the system of equations.
Ex 1:
Write an augmented matrix for each system of equations. Then solve with a graphing calculator.
4 y  6 x  10
2 x  7 y  22
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AMHS Precalculus - Unit 5 extension
Ex 2:
Write an augmented matrix for each system of equations. Then solve with a graphing calculator.
3 x  5 y  2 z  7
4 x  3 y  5 z  19
5 x  4 y  7 z  15
Application: Matrices and Systems of Equations
Ex 1: The admission fee for the Coastal Carolina Fair is $1.50 for children and $4.00 for adults. On a
certain day, 2200 people enter the fair and $5050 is collected. How many children and how many adults
attended?
Ex 2: A cashier has 25 coins consisting of nickels, dimes, and quarters with a value of $4.90. If the
number of dimes is 1 less than twice the number of nickels, how many of each type of coin does she
have?
Ex 3: Sully, Krista, and Steve are combining their savings for a flight to Cancun. Together, they have
$680. Sully has saved $20 less than Krista. Steve has managed to triple Krista and Sully’s savings
combined. How much has each contributed to the total savings?
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AMHS Precalculus - Unit 5 extension
Systems of Equations without a Unique Solution
An Inconsistent System
3x  2 y  4
6 x  4 y  7
Ex 1:
A System with Infinitely Many Solutions
9 x  3 y  24
3x  y  8
2 x  3z  3
Solve the system using the graphing calculator. 4 x  3 y  7 z  5
8 x  9 y  15 z  9
3x  2 y  4 z  1
Ex 2: Solve the system using the graphing calculator. x  y  2 z  3
2x  3y  6z  8