IB Math SL - Santowski1 Lesson 54 – Multiplication of Matrices Math 2 Honors - Santowski.

IB Math SL - Santowski 1

Lesson 54 – Multiplication of Matrices

Math 2 Honors - Santowski

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



(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

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



(F) Properties of Matrix Addition


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

Multiplying Matrices - Generalized Example If we multiply a 2×3 matrix with a 3×1 matrix, the product

matrix is 2×1

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


21

11

31

21

11

232221

131211

M

M

t

t

t

rrr

rrr


(B) Matrix Multiplication - Summary Summary of Multiplication process


(D) Examples for Practice

Multiply the following matrices:

642

432

321

1-10

1-01

001

(f)

011

203

024

122

213 (e)

31

1-2

10

11 (d)

011

203

024

2-1-3 (c)

10

11

31

1-2 (b)

002

110

011

751

1102 (a)


(D) Examples for Practice

Multiply the following matrices:

u

t

s

(b)

dc

ba (a)

zw

yv

xu

fed

cba

zy

xw


(E) Examples for Practice – TI-84 Here are the key steps involved in using the TI-84

10

11

31

1-2 (a)


(E) Examples for Practice – TI-84 Here are the key steps involved in using the TI-84

011

203

024

122

213 (a)


(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???


(F) Properties of Matrix Multiplication


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

a x 1 = 1 x a = a

an exists for all a > 0


(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.


(C) Key Terms for Matrices

Definition An identity matrix is a square matrix with ones along the main diagonal and zeros elsewhere.

So, in matrix multiplication A x I = I x A = A

1000

0100

0010

0001

100

010

001

10

01I


(F) Properties of Matrix Multiplication


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

a x 1 = 1 x a = 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


(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 ?

(2) In general, does AB = BA? (3) Does A(BC) = (AB)C? (4) Does A(B + C) = AB + AC?

a

aA

0

0


(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?

Internet Links

http://www.intmath.com/matrices-determinants/3-matrices.php

http://www.purplemath.com/modules/mtrxadd.htm


IB Math SL - Santowski1 Lesson 54 – Multiplication of Matrices Math 2 Honors - Santowski.

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