Download 3-5 Perform Basic Matrix Operations

3-5 Perform Basic Matrix Operations
Name:_____________________
Objective: To perform basic operations with matrices.
Algebra 2 Standard 2.0
*A matrix is a rectangular arrangement of numbers in __________ and ___________________.
4  1 5 
Ex. 

0 6 3
*Two matrices are equal if their dimensions are the same and the elements in the corresponding
positions are equal.
*To add or subtract two matrices, simply add or subtract corresponding elements.
You can add or subtract matrices only if they have the _____________ dimensions.
a b  e


c d g
f  ae b f 


h  c  g d  h
a b   e
Subtracting Matrices 

c d  g
f   a e b f 


h  c  g d  h
Adding Matrices
* Perform the indicated operation, if possible.
Ex. 1:
 2 4   3 2 
a. 


0 1  4 0
You Try:
 2 11   3 1 
a. 


 4 5   2 8 
 4 6  2 3
b.  2 2    4 2 
1
5   3 1 

2 
 4 0   2




b.  7
2    3 0 
 3 1   5 14 

 

*Scalar Multiplication:
Ex. 2: Perform the indicated operation, if possible.
a.
 3 2 
4  0 3 
1 6 


 1 5   2 1 
b. 3 


 4 0   5 3 
You Try:
 2 1 3 


a. 4  7 6 1 
 2 0 5 


Algebra 2 Ch.3B Notes Page 1
 4 1   2 2 
b. 3 


 3 5   0 6 
*Properties of Matrix Operations:
Let A, B, and C be matrices with the same dimensions, and let k be a scaler.
_____________________Property of Addition:  A  B   C  A   B  C 
_____________________Property of Addition: A  B  B  A
_____________________Property of Addition: k  A  B   kA  kB
_____________________Property of Subtraction: k  A  B   kA  kB
Ex. 3; A local bakery keeps track of their sales as shown below.
Last Month: Store 1: 650 rolls, 220 cakes, 32 pies; Store 2: 540 rolls, 200 cakes, 30 pies
This Month: Store 1: 840 rolls, 250 cakes, 50 pies; Store 2: 800 rolls, 250 cakes, 42 pies
Organize the data using matrices. Then write and interpret a matrix giving total number of
bakery items sold per store.
Ex. 4: Solve the matrix equation for x ad y.
 2x 1   2 3   4 4 




 4 y   1 4   3 6 
Ex. 5: Solve the matrix equation for x ad y.
  2 x 3   1 4   10 2 
2

  

  5  y   3 5   16 14 
You Try: Solve the matrix equation for x ad y.
  3x 1  9 4   12 10 
2  

  

y   5 3    2 18 
 4
Algebra 2 Ch.3B Notes Page 2
3-6 Multiply Matrices
Name:____________________
Objective: To multiply matrices.
Algebra 2 Standard 2.0
* The product of two matrices A and B is defined (possible) when
the number of __________ in A is ________________ to the number of _____________ in B.
 A   B   AB
( m  n) ( n  p )  ( m  p )
Ex. 1: State whether the product is defined. If so, give the dimensions of AB.
a.
A: 3×5, B: 5×2
b.
A: 3×4, B: 3×2
You Try: State whether the product is defined. If so, give the dimensions of AB.
a.
A: 5×2, B: 2×2
b.
A: 3×2, B: 3×2
A

*Multiplying Matrices:  a b   e


c
d

 g
B

AB
f   ae  bg

h   ce  dg
af  bh 

cf  dh 
 2 3 
 1 4 
Ex. 2: Find AB if A = 
 and B = 
.
1 5 
 3 2 
 3 3 
You Try: Find AB if A = 
 and B =
 1 2 
Algebra 2 Ch.3B Notes Page 3
1 5

.
 3 2 
* Using the given matrices, evaluate the expression.
 3 2 
 2 3 
 2 1

A   0
4  ; B
 ; C 

1 0 
 4 2 
 1 5 


Ex. 3: A(B-C)
You Try: A(B+C)
*Properties of Matrix Multiplication:
Let A, B, and C be matrices and k be a scalar.
Associative Property of Matrix Multiplication: ________________________________
Left Distributive Property :________________________________________________
Right Distributive Property :_______________________________________________
Associative Property of Scalar Multiplication: _________________________________
 Inventory  Cost per item  Total Cost 

   matrix

Cost Matrices:  matrix   matrix
 

m n
n p
m p
Ex. 4: The matrix below represents the inventory for a chain of entertainment stores. The first
row is Store 1, the second row is Store 2, and the third row is Store 3.
CDs DVD VHS Games
 2800 550 200 150 


 2600 800 150 120 
 1850 650 190 100 


If CD’s cost $15, DVDs cost $20, VHSs cost $18, and Games cost $30, find the total value of the
inventory for each store.
Algebra 2 Ch.3B Notes Page 4
3-7 Evaluate Determinants and Apply Cramer’s Rule
Objective: To evaluate determinants of matrices.
Algebra 2 Standard 2.0
Name:____________________
*Determinant: A real number associated with any square (n×n) matrix.
The determinant of a matrix A is denoted by _____________ or by _____________.
a b a b
*Determinant of a 2×2 Matrix: det 
 ad  cb

c d c d
*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


Ex 1: Evaluate the determinants:
 6 2 
a. 

 1 4 
 4 2 0 


b.  1 1 2 
2 5 3 


10 2 3 


c.  2 12 4 
 0 7 2 


You Try: Evaluate the determinants:
 4 1 2 
 3 2 


a. 
b.  3 2 1 

6 1 
 0 5 1


*Area of a Triangle:
The area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is given by
x1
1
Area   x2
2
x3
y1 1
y2 1
y3 1
where the symbol ± indicates that the appropriate sign should be chosen to yield a positive value.
Algebra 2 Ch.3B Notes Page 5
Ex. 2: The approximate coordinates (in miles) of a triangular region representing a city and its
suburbs are (10, 20), (-8, 5) and (-4, -5). Find the area of the region.
10 20 1
1
Area   8 5 1
2
4 5 1
You Try: Find the area of the triangle shown.
* Cramer’s Rule for 2x2 system
Let A be the coefficient matrix of this linear system:
ax + by = e
cx + dy = f
If det A  ___, then the system has ___________ solution.
a e
e b
c f
f d
y = ________
x = _______and
det A
det A
Use Cramer’s rule to solve the system.
Ex. 3: 3x + 2y = 4
2x  7y = 11
You Try: 3x + y = 4
5x + 4y =  5
Algebra 2 Ch.3B Notes Page 6
* Cramer’s Rule for 3x3 system
Let A be the coefficient matrix of the linear system:
ax + by + cz = j
dx + ey + fz = k
gx + hy + iz = l
If det A  ____, then the system has ________________solution.
j b c
a j c
k e f
d k f
l h i
g l i
x=____________,
y= __________,
det A
det A
Use Cramer’s rule to solve the system.
Ex. 4:
3x + 4y  z = 9
2x  3y + 4z = 14
4x  y + z = 18
You Try:
2x  3y 2z = 10
x + 2y + 3z = 14
4x + y + 2z = 4
Algebra 2 Ch.3B Notes Page 7
a b j
d e k
g h l
z = _________
det A
3-8A Use Inverse Matrices to Solve Linear Systems
Name:____________
Objective: To find the inverse of 2×2 matrices.
Algebra 2 Standard 2.0
* The n×n identity matrix is a matrix with ____ on the main diagonal and ____ elsewhere.
If A is any n×n matrix and I is the n×n ___________ matrix, then AI =A and IA=A.
2×2 Identity Matrix
1 0
I 

0 1
3×3 Identity Matrix
1 0 0


I   0 1 0
0 0 1


* Two n×n matrices A and B are inverse of each other if their __________ (in both orders) is the
n×n ________________ matrix. That is, AB = I and BA = I.
An n×n matrix has an inverse if and only if det A ≠ 0. The symbol for the inverse of A is A-1.
The inverse of a 2×2 matrix:
a b
The inverse of the matrix A  
 is
c d 
A1 
1  d b 
1  d b 




A  c a  ad  bc  c a 
provided ad  bc  0
 5 2 
Ex. 1a: Find the inverse of A  

 8 3 
6 1
Ex. 1b: Find the inverse of A  

 2 4
 1 5 
You Try: Find the inverse of A  

 4 8 
Algebra 2 Ch.3B Notes Page 8
*Solve a matrix equation:
1. Find the inverse matrix of A.
2. Multiply each side of AX = B by A-1 on the ____________ to find the solution X = A-1B.
Ex. 2a: Solve the matrix equation AX = B for the 2×2 matrix X.
 3 2 
 2 4 

X 

 7 5 
 3 1
Ex. 2b: Solve the matrix equation AX = B for the 2×3 matrix X.
 5 2 
 4 5 0

X 

 9 3 
3 1 6
You Try: Solve the matrix equation AX = B for the 2×2 matrix X.
 4 1 
 8 9

X 

 0 6
 24 6 
Algebra 2 Ch.3B Notes Page 9
3-8B Use Inverse Matrices to Solve Linear Systems
Name________________
Objective: To solve linear systems using inverses.
Algebra 2 Standard 2.0
* The inverse of a 3×3 matrix is very difficult to compute by hand, so we will use a calculator.
* Use a graphing calculator to find the inverse of A. Then use the calculator to verify the result
by showing that AA-1 = I and A-1A =I
Ex. 1:
Ex. 2:
1

A 3

 1
2

A 2

 12
You Try:
0
2 
2
1
1
4
2


2

6 
0
0
4
2

A 5

4



1
2 
3
0
3
8



*Using Inverse Matrices to Solve Linear Systems:
3. Write the system as a matrix equation Ax = B. The matrix A is the coefficient matrix, X
is the matrix of variables, and B is the matrix of constants.
4. Find the inverse matrix of A.
5. Multiply each side of AX = B by A-1 on the ___________ to find the solution X = A-1B.
Ex. 3: Use an inverse matrix to solve the linear system.
5x – 2y = -10
2x – 4y = 12
Algebra 2 Ch.3B Notes Page 10
Ex. 4: Use an inverse matrix to solve the linear system.
2 x  y  6 z  4
6 x  4 y  5 z  7
4 x  2 y  5 z  9
You Try: Use an inverse matrix to solve the linear system.
3 x  y  2 z  10
6 x  2 y  z  2
x  4 y  3z  7
Ex. 5: At a video store, the cost of 3 DVDs, 2 video games, and 4 VHS movies is $130. The
cost of 2 DVDs, 1 video game, and 5 VHS movies is $105. The cost of 3 DVDs, 3 video games,
and 3 VHS movies is $135. Find the cost of each item.
Algebra 2 Ch.3B Notes Page 11