Transcript Link to ppt Lesson Notes

Lesson 54 – Multiplication of
Matrices
Math 2 Honors - Santowski
IB Math SL - Santowski
1
Lesson Objectives




(1) Review simple terminology associated
with matrices
(2) Review simple operations with matrices
(+,-, scalar multiplication)
(3) Compare properties of numbers with
matrices (and at the same time introduce the
use of the GDC)
(4) Multiply matrices
IB Math SL - Santowski
2
(F) Properties of Matrix Addition
Ordinary Algebra with Real
numbers
Matrix Algebra
if a and b are real numbers, so is the
product a+b
Commutative:
a + b = b + a for all a,b
Associative:
(a + b) + c = a + (b + c)
Additive Identity:
a + 0 = 0 + a = a for all a
Additive Inverse:
a + (-a) = (-a) + a = 0
IB Math SL - Santowski
3
TI-84 and Matrices

Here are the screen captures on HOW to use
the TI-84 wherein we test our properties of
matrix addition

Use 2nd x-1 to access the matrix menu
IB Math SL - Santowski
4
(F) Properties of Matrix Addition
Ordinary Algebra with Real
numbers
Matrix Algebra
if a and b are real numbers, so is the
product a+b
if A and B are matrices, so is the sum A + B
provided that …..
Commutative:
a + b = b + a for all a,b
in general, A + B = B + A provided that …..
Associative:
(a + b) + c = a + (b + c)
(A + B) + C = A + (B + C) is true provided
that …..
Additive Identity:
a + 0 = 0 + a = a for all a
A + 0 = 0 + A = A for all A where 0 is the
zero matrix
Additive Inverse:
a + (-a) = (-a) + a = 0
A + (-A) = (-A) + A = 0
IB Math SL - Santowski
5
Multiplying Matrices - Generalized Example

If we multiply a 2×3 matrix with a 3×1 matrix, the product
matrix is 2×1
 r11 r12
r r
 21 22



t11 
r13     M 11 
 t 21   


r23 
M
t31   21 
Here is how we get M11 and M22 in the product.
M11 = r11× t11 + r12× t21 + r13×t31
M12 = r21× t11 + r22× t21 + r23×t31
IB Math SL - Santowski
6
(B) Matrix Multiplication - Summary

Summary of Multiplication process
IB Math SL - Santowski
7
(D) Examples for Practice

Multiply the following matrices:
1 5  7 
1 1

0

(a) 2 0  1 1 
0  1 1 


2
0
0


4 2 0
(c) 3 - 1 - 2 3 0 2
1 1 0
4 2 0
 3  1  2 

(e) 

3
0
2
 

2

2

1

 1 1 0 


2 - 1 1 1 
(b) 



1
3
0

1

 

1 1  2 - 1
(d) 



0

1
1
3

 

1 0 0  1 2 3
(f) 1 0 - 1  2 3 4
0 1 - 1 2 4 6
IB Math SL - Santowski
8
(D) Examples for Practice

Multiply the following matrices:
a b   w x 
(a) 



c d   y z 
a b
(b) 
d e
u
c 
 v

f
 w
IB Math SL - Santowski
x s

y t
z u 
9
(E) Examples for Practice – TI-84

Here are the key steps involved in using the TI-84
IB Math SL - Santowski
2 - 1 1 1 
(a) 



1 3  0  1
10
(E) Examples for Practice – TI-84

Here are the key steps involved in using the TI-84
IB Math SL - Santowski
4 2 0
 3  1  2 
(a) 
 3 0 2

 2  2  1  1 1 0 


11
(F) Properties of Matrix Multiplication

Now we pass from the concrete to the
abstract  What properties are true of matrix
multiplication where we simply have a matrix
(wherein we know or don’t know what
elements are within)

Asked in an alternative sense  what are the
general properties of multiplication (say of
real numbers) in the first place???
IB Math SL - Santowski
12
(F) Properties of Matrix Multiplication
Ordinary Algebra with Real
numbers
Matrix Algebra
if a and b are real numbers, so is the
product ab
ab = ba for all a,b
a0 = 0a = 0 for all a
a(b + c) = ab + ac
ax1=1xa=a
an exists for all a > 0
IB Math SL - Santowski
13
(C) Key Terms for Matrices

We learned in the last lesson that there is a matrix version of the
addition property of zero.

There is also a matrix version of the multiplication property of one.

The real number version tells us that if a is a real number, then a*1 =
1*a = a.

The matrix version of this property states that if A is a square matrix,
then A*I = I*A = A, where I is the identity matrix of the same
dimensions as A.

Definition  An identity matrix is a square matrix with ones along
the main diagonal and zeros elsewhere.
IB Math SL - Santowski
14
(C) Key Terms for Matrices

Definition  An identity matrix is a square matrix with ones along
the main diagonal and zeros elsewhere.
1
1 0 0 
0
1 0 


I 
 0 1 0  

0 1  0 0 1  0

 0


0 0 0
1 0 0
0 1 0

0 0 1
So, in matrix multiplication  A x I = I x A = A
IB Math SL - Santowski
15
(F) Properties of Matrix Multiplication
Ordinary Algebra with Real
numbers
Matrix Algebra
if a and b are real numbers, so is the
product ab
if A and B are matrices, so is the product
AB
ab = ba for all a,b
in general, AB ≠ BA
a0 = 0a = 0 for all a
A0 = 0A = 0 for all A where 0 is the zero
matrix
a(b + c) = ab + ac
A(B + C) = AB + AC
ax1=1xa=a
AI = IA = A where I is called an identity
matrix and A is a square matrix
an exists for all a > 0
An for {n E I | n > 2} and A is a square
matrix
IB Math SL - Santowski
16
(F) Properties of Matrix Multiplication

This is a good place to use your calculator if it
handles matrices. Do enough examples of each to
convince yourself of your answer to each question

(1) Does AB = BA for all B for which matrix
multiplication is defined if
?
a 0 
A

0
a





(2) In general, does AB = BA?
(3) Does A(BC) = (AB)C?
(4) Does A(B + C) = AB + AC?
IB Math SL - Santowski
17
(F) Properties of Matrix Multiplication

This is a good place to use your calculator if it
handles matrices. Do enough examples of each to
convince yourself of your answer to each question

(6) Does A - B = -(B - A)?

(7) For real numbers, if ab = 0, we know that either a
or b must be zero. Is it true that AB = 0 implies that
A or B is a zero matrix?
IB Math SL - Santowski
18
Internet Links

http://www.intmath.com/matricesdeterminants/3-matrices.php

http://www.purplemath.com/modules/mtrxadd
.htm
IB Math SL - Santowski
19