Transcript Matrices - Northside Middle School

Matrices
IB Mathematics SL
Matrices
Describing Matrices
Adding Matrices
Matrix (Plural: Matrices)


A matrix is a rectangular array of numbers
used to store information
Terminology:




Dimensions or “Order” (#rows x #columns)
Entries
Square matrices, Vectors
Future Application: Matrices are tools that
can be used to analyze multiple
dimensions and multiple variables (not just
2 and 3)
Row and Column Vectors


A matrix with one dimension equal to 1 is called a
vector
Below is a “column vector”
3 


Example :  0 
 10 


Adding Matrices


To add matrices, order must be identical
Add corresponding entries to create a new
matrix of the same order
Add the Following Matrices
1 0.5   0
1
7 
 6

 

7
0
0

80
100

100


 

 10 0.75 10   5 .5

6 

 
 4
  3
5




  4
6  
 
Matrices
Multiplication of Matrices
Today



Review Addition of Matrices
Multiplication of Matrices
Identity Matrices, Determinants [and Inverse
Matrices]
To Multiply Matrices by a
Scalar (a number)

Multiply each entry by the same scalar
11 0.1

  33 0.3 
2




3
1   2 3 
3
 

10
0

30
0
 



 2 5 
1

3

2
56 
 4

Matrix Multiplication Activity

Handout 2: Flying Matrices


Advice: Solve the problem without matrices,
putting your answer in a matrix at the end
Goal


Create Matrices
Multiply Matrices
To Multiply a Matrix by a Matrix

The number of COLUMNS in the first matrix must
match the number of ROWS in the second matrix

Take a look at your Flying Matrices problem. Notice the
location of the Feed and Calc. labels:
W V
Feed Calc
 500
200 
Feed  40 2 


Calc  50 3 
To Multiply a Matrix by a Matrix

What was done with the (500 200) row and the (40
50) column to get 30,000?
Feed Calc
W V
W
V
 500 200  Feed  40 2 



  Mon  30, 000 1600 

 Calc  50 3 
Multiplication of Matrices


The number of Columns in the first matrix
must match the number of ROWS in the
second matrix
Multiply entire ROWS in the first matrix by
entire COLUMNS in the second matrix
Multiplying a 2x2 by a 2x2
M 500 200   40



T  400 300  50
M 500 40  200 50
T  400 40  300 50
M 30, 000 1600 
T  31, 000 1700 
2


3
500 2  200 3


400 2  300 3
Practice Multiplication of
Matrices
11 12 

 a b  

 21 23
11 12 

1 2  

 21 23
10 15  1 2 
 6 9    1 2  

 

5 0   6 6 
0 5    3 3 

 

11 12 

a b c  

 21 23
 5 9 
 0 3 
1
2
3


 

 5 1 
 2 1
1 2 3  



1
0
0 2 4  


 0 1


Determining the Dimensions of
the Product



Write out the dimensions of the two matrices
 (1x2)(2x2)
 (2x3)(3x2)
If the two middle numbers line up, then
multiplication is possible (the columns in the
first matrix match the rows in the second
matrix)
The two outer numbers give the dimensions
of the product
 (1x2)(2x2)(1x2)
 (2x3)(3x2)(2x2)
Calculators
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
Go to the “Matrix” Menu and scroll to “Edit”
Select Matrix [A]: Enter the dimensions and
the entries of the first matrix, Quit
Select Matrix [B]: Enter the dimensions and
the entries of the first matrix, Quit
Calculate [A][B]
Matrix Topics


NO Commutative Property of Multiplication
Identity Matrix (for square matrices)


Square matrix with all zero entries except for the
top left to bottom right diagonal, which contains all
ones
Determinant


We will analyze square matrices only
We will look at 2x2 matrices first, and 3x3
matrices later
Properties of the Identity

Let A be a square matrix of order NxN
“I” is the Identity Matrix of order NxN

The following statement is true

IA A IA
The Determinant

The determinant “determines” whether or not a
matrix is “non-singular” or “invertible”.



If the determinant is non-zero, the matrix is non-singular
If the determinant is zero, the matrix is singular, and has no
inverse
The Inverse, and the ability to calculate an inverse
will be important when we use matrices to solve
systems of equations in multiple variables (this is a
common application of matrices)
a b 
Determinant  

ad

bc

c d 
Activity
Matrix Multiplication
 Exercise 11.1.2:

3 and 9
The Determinant
 Exercise 11.2, Problem 1

v, ix, x, xi
Matrices
Inverse Matrices
Today

Review:




Matrix Operations
The Identity and the Determinant
Finding the Inverse Matrix A-1
Calculator approaches
Matrix Operations

Addition



Multiplication by a Scalar


To add matrices, order must be identical
Add corresponding entries to create a new matrix
of the same order
Multiply each entry by the same scalar
Multiplication by a Matrix



The number of Columns in the first matrix must
match the number of ROWS in the second matrix
Multiply entire ROWS in the first matrix by entire
COLUMNS in the second matrix
Note: No Commutative Property
Determinant, Identity, Inverse
of Square Matrices
a b 
Determinant  
 ad  bc

c d 
Identity : AI  A  IA
1
Inverse : A A  I  AA
1
The Inverse

For matrix A, the Inverse matrix “A-1” is such that the
product of the matrix and its inverse is the Identity
matrix
1
1
Inverse : A A  I  AA
Calculating the Inverse Matrix
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
Derivation (on 388-389)
Though you will not be required to “re-derive” the
inverse, you will have to calculate the inverse matrix
for 2x2 matrices
a b 
1  d b 
1
if : A  
A 



A  c a 
c d 
b 
 d
1  d b   ad  bc ad  bc 





a 
ad  bc  c a   c
 ad  bc ad  bc 
The Determinant

Note the position of the determinant in the
denominator. If the determinant is equal to
ZERO, then the Inverse is not defined (or
“there is no Inverse”).


If the determinant of a matrix is equal to 0, the
matrix is called “singular”
Otherwise, it is “non-singular”
Determining Determinants
(requires Determination)

Task 1: Exercise 11.2,



Problem 2: ii, vi, x, xiv
Problems 3 and 5
Task 2: Find the determinant of a 2x2 Identity
matrix
Matrices
2x2 and 3x3 Inverse Matrices in
the Calculator
Solving Systems of Equations
with Matrices
Calculating the Inverse Matrix


Derivation (on 388-389)
Though you will not be required to “re-derive” the
inverse, you will have to calculate the inverse matrix
for 2x2 matrices
a b 
1  d b 
1
if : A  
A 



A  c a 
c d 
b 
 d
1  d b   ad  bc ad  bc 





a 
ad  bc  c a   c
 ad  bc ad  bc 
The Determinant

Note the position of the determinant in the
denominator. If the determinant is equal to
ZERO, then the Inverse is not defined (“there
is no Inverse”).


If the determinant of a matrix is equal to 0, the
matrix is called “singular”
Otherwise, it is “non-singular”
Calculators


Having created a square matrix with your
Matrix option, you can…
Find a Determinant


Find the Inverse Matrix


Matrix  Math  ”det”
Calculate [A]-1
Activity: Check your answers for 2-ii and
2-vi from the homework
Determinants for 3x3 Matrices
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
You will be expected to calculate the inverse
of a 2x2 matrix by hand. The formula is in
your formula packet.
3x3 inverses may be found with a calculator


Create the 3x3 matrix in your Matrix menu, then
use the “-1” button to find the inverse.
Note: The determinant can be obtained through
the MATH menu. Remember: if the determinant
is zero, the matrix is singular and has NO inverse.
Activity

Exercise 11.2

Use your calculator to answer Problem 6 and 7, a
and b on each.
Systems of Equations
(or “Simultaneous Equations”)


Goal:
We will use 2x2 and 3x3 matrices to solve
systems of equations in 2 or 3 variables
Systems of Equations
(or “Simultaneous Equations”)


Systems of equations
can be written using
matrices
An example of a 2variable system is on
the right
4 x  2 y  2
4 x  y  11
 4 2   x   2 
 4 1  y    11

  

Systems of Equations
(or “Simultaneous Equations”)


Systems of equations
can be written using
matrices
An example of a 3variable system is on
the right
x  2y  z  6
x  y  z  1
3x  y  2 z  2
1 2 1   x   6 
1 1 1  y    1

   
3 1 2   z   2 
Practice


Write each of the
systems on the right in
matrix form
Then, solve the system
using your calculator
3 x  y  19
4 x  y  23
2 x  3 y  44
y  3 x
2 x  2 y  z  8
4x  2 y  2z  2
2 x  2 y  3 z  16
Solving Systems of Equations


You can use “Substitution” or “Gaussian
Elimination” to solve systems of equations
Matrices can be used to simplify the process

3-variable elimination is fairly complicated, but
using eliminations in a system with 4, 5 or more
variables is EXTREMELY tedious
Solving Systems of Equations


Let’s go back to the first
example of a twovariable system
In the last line, I have
simplified matrix form
by defining the matrices
A and B

Note: this is not
necessary, but the slide
would be very busy if I
didn’t do this
4 x  2 y  2
4 x  y  11
 4 2   x   2 
 4 1  y    11

  

x 
 A      B 
 y
Note :
Solving for x and y  A1  A  I


Since the product of A
and its inverse is the
identity, the column
vector containing x and
y can be isolated
The column vector
containing x and y can
be obtained by
evaluating A-1B
x  x 
I  
 y  y
Solution :
x 
 A     B 
 y
x 
1
1
 A  A      A   B 
 y
x 
1
 y    A  B 
 
Solution
4 x  2 y  2
4 x  y  11
 4 2   x   2 
 4 1  y    11

  

1
 x   4 2   2   2 
 y    4 1  11  3 
  
 
  
x  2, y  3
Summary
4 x  2 y  2



Write the system in
matrix form
Calculate the product or
A-1 and B
Give your solutions
4 x  y  11
 4 2   x   2 
 4 1  y    11

  

1
 x   4 2   2   2 
 y    4 1  11  3 
  
 
  
x  2, y  3
Practice

Solve each of the
systems on the right
using matrices
3 x  y  19
4 x  y  23
2 x  3 y  44
y  3 x
2 x  2 y  z  8
4x  2 y  2z  2
2 x  2 y  3 z  16
Challenge: Solve this 6variable system!
a  b  c  d  e  f  30
2a  3b  6c  4d  e  f  8
5a  4b  3c  d  5e  2 f  34
2a  3b  8c  6d  e  4 f  38
6a  2b  7c  5d  3e  2 f  42
5a  8b  5c  3d  9e  4 f  18
Other Concepts

For a system of n variables, you need n
equations

The coefficient matrix [A] needs to be a square

What if the coefficient matrix is singular?

These problems can all be solved using
Gaussian elimination…but it will be fairly
time-consuming
Homework

Exercise 11.3



Problem 1 (iii, vi, ix)
Problem 4 (a, c)
Next Class, Quiz on:

Matrices



Addition, Multiplication
Determinants and the Inverse
Solving 2- and 3-variable systems
Quiz Details

Matrices


Addition, Multiplication
Determinants and the Inverse



Singular vs. Non-singular
You should be able to find the determinant and inverse
of a 2x2 matrix by hand. For a 3x3 matrix, you may
use a calculator
Solving 2- and 3-variable systems

WITH a calculator