© 2005 Baylor University Slide 1 Fundamentals of Engineering Analysis EGR 1302 - Matrix...

© 2005 Baylor UniversitySlide 1

Fundamentals of Engineering AnalysisEGR 1302 - Matrix Multiplication, Types


Matrix Multiplication

Define “Conformable”

To multiply A * B, the matrices must be conformable.

Given matrices: A m x n and B n x p

The number of “Columns” n of A, must equal the number of “Rows” n of BWhich defines the order of the multiplication

A=

2 x 3

For A: m=2, n=3 B=

3 x 4

For B: n=3, p=4

Note that B * A is Undefined (not allowed) because p = m

For A * B, n=n; i.e. 3=3, so A*B is “conformable”


Order of Multiplication

The order in which a multiplication is expressed is important.

We use the terms “pre-multiply” or “post-multiply” to stipulate the order.

Given A * B = C, we say that “B” is “pre-multiplied” by “A”(we could also say that A is post-multiplied by B).

In other words, Matrix Multiplication is NOT Commutative(except in special cases)

Because matrices must be conformable for multiplication; in general

A * B = B * A



232221

131211

aaa

aaaA

m x n2 x 3

232221

131211

cca

cccC

m x p

333231

232221

131211

bbb

bbb

bbb

B

n x p3 x 3

=

A * B = C is Conformable

Is a Row on Column operation

The Product C will be a 2 x 3



232221

131211

aaa

aaa

333231

232221

131211

bbb

bbb

bbb

232221

131211

cca

ccc* =

31132112111111 bababac

C11 is made up of Row 1 from A, and Column 1 from B

Note the “sum of products” form

C12 is made up of Row 1 from A, and Column 2 from B

32132212121112 bababac

Remember: 31132112111111 bababac



232221

131211

aaa

aaaA

3231

2221

1211

bb

bb

bb

B

A * B =

333231

232221

131211

ccc

ccc

ccc

C

2221

1211

cc

ccC

B * A = 3 x 3

2 x 2


112

310

23

10

21

* =9 5

1 7


A * B =



4 3 -1

2 1 1

4 -1 11

112

310

23

10

21

B * A =

* =

Work this out yourself, before proceeding,

To make sure you understand the method of matrix multiplication.


Linear Systems as Sum of Products

ax1 + bx2 + cx3 = d

Sum of Products form

[ a b c ] - a 1 x 3 row vector

x1

x2

x3

- a 3 x 1 column vector

[ a b c ] * = [ d ] - a 1 x 1 scalar – i.e.;

x1

x2

x3

ax1 + bx2 + cx3 = d


Conformability and Order of Matrix Multiplication

Given: A5x4 B4x5 C6x4

A * B = D5x5

B * A = E4x4

A * C = not conformable

C * A = not conformable

C * B = F6x5

A * B * C = not conformable

C * B * A = G6x4


Properties of a Zero Matrix

123

011

21

21

21

* =

00

00

In Algebra, x * 0 = 0, but if x = 0, and y = 0, then x * y = 0

In Matrix Algebra, even if A = 0, and B = 0, A * B can be [0]

Note that:

123

011

21

21

21* =

257

257

257


Matrix Form of Linear Equations

Distributive Property: A(B+C) = AB + AC

Associative Property: A(BC) = (AB)C

111 cba

3

2

1

x

x

x 312111 xcxbxa

1312111 dxcxbxa

2322212 dxcxbxa

3332313 dxcxbxa

Then can become

333

222

111

cba

cba

cba

3

2

1

x

x

x

3

2

1

d

d

d

A x

d

* = Any Order

?How do wesolve thissystem

of equations


Special Matrices

The Transpose Matrix

Rule: The Row becomes the Column, and the Column becomes the Row

232221

131211

aaa

aaaA

2313

2212

2111

aa

aa

aa

AT

A is a 2x3, so AT will be a 3x2

333231

232221

131211

bbb

bbb

bbb

B

332313

322212

312111

bbb

bbb

bbb

BTFor a 3x3


Properties of the Transpose Matrix

112

101TA

11

10

21

A

221

013B

20

21

13TB

11

10

21

221

013

232

221

435

A*B=

AT*BT = ?

BT*AT =

20

21

13

112

101

224

323

215

(AB)T= BT*AT


Additional Properties of the Transpose

If A+B and A*B are allowed (are conformable), then

(A+B)T = AT + BT

(AB)T = BTAT


The Symmetric Matrix

212

123

235

A = AT

Must be Square: n x n

jiij aa

A + AT must also be Symmetric

The Diagonal


The Diagonal Matrix

300

020

005

Must be Square: n x nAll off-diagonal elements

Are Zero

333322221111 ,, bababa

If A and B areDiagonal

200

040

001

300

020

005

+A+B

will be Diagonal

500

060

006

=

200

040

001

300

020

005If A and B areDiagonal * A*B

will be Diagonal

600

080

005

=


The Identity Matrix

100

010

001Must be Square: n x nAnd must be Diagonal

Can be any Order Notation: IN The Unity term

A*I = AI*A = A

A does not have to be square

Amxn * In = A or Im * Amxn = A


Powers of Matrices

A * A = A2 for Square Matrices Only

A * A2 = A3 … and so on

If A is Diagonal … A2 = a112, a22

2, a332

200

010

003

200

010

003

400

010

009

=*


Matrix Math on the TI-89 Calculator

My Philosophy for using Calculators(and Computers …)

Be aware of theOrder of Magnitude

Sign Errors are easy to miss

Double check your work

If you understand the solution methodology,You will understand the answer.


103

231

012

A

412

150B

A*B – not conformable B*A = ?

Matrix Math on the TI-89 Calculator


412

150B

Matrix Math on the TI-89 Calculator (cont.)


Matrix Math on the TI-89 Calculator (cont.)


Using the Matrix Editor on the TI-89


Questions?

© 2005 Baylor University Slide 1 Fundamentals of Engineering Analysis EGR 1302 - Matrix...

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