Download Matrix Algebra

Matrix Algebra
Rate of Change
Rate of temperature change, degrees/hour
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Chapter 10
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time, hour
Applied Force
F = 200 lbf
20 feet
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θ = 600
500
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Pivot Point
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
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10.1 Matrix Operations
and Functions
Rate of Change
Rate of temperature change, degrees/hour
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• Matrix algebra is used extensively
in engineering applications
• Matrix algebra is different from the
array calculations we have
performed thus far
400
500
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200
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500
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
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Array Operators
Rate of Change
Rate of temperature change, degrees/hour
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• A.* B
time, hour
• A./B
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500
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• A.^B
multiplies each element in
array A times the
corresponding element in
array B
divides each element in
array A by the corresponding
element in array B
raises each element in array
A to the power in the
corresponding element of
array B
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
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transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
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Operators used in Matrix
Mathematics
Rate of Change
Rate of temperature change, degrees/hour
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•
•
•
•
•
Transpose
Multiplication
Division
Exponentiation
Left Division
500
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
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transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
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Some Matrix Algebra
functions
Rate of Change
Rate of temperature change, degrees/hour
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time, hour
100
200
•
•
•
•
Dot products
Cross products
Inverse
Determinants
300
400
500
100
200
300
400
500
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
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Transpose
Rate of Change
Rate of temperature change, degrees/hour
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time, hour
• In mathematics texts you will
often see the transpose indicated
with superscript T
• AT
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500
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200
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• The MATLAB syntax for the
transpose is
• A'
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
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The transpose switches the
rows and columns
Rate of Change
Rate of temperature change, degrees/hour
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time, hour
1 2 3
4 5 6

A
7 8 9


10 11 12
100
200
300
1 4 7 10
AT  2 5 8 11
3 6 9 12
400
500
100
200
300
400
500
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
Rate of Change
Rate of temperature change, degrees/hour
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time, hour
100
200
300
400
500
100
200
300
400
500
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
Using the transpose with
complex numbers
Rate of Change
Rate of temperature change, degrees/hour
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time, hour
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When used with complex
numbers, the transpose
operator returns the complex
conjugate
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
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Dot Products
Rate of Change
Rate of temperature change, degrees/hour
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4
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time, hour
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200
300
• The dot product is sometimes
called the scalar product
• the sum of the results when you
multiply two vectors together,
element by element.
400
500
100
200
300
400
500
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
Rate of Change
Rate of temperature change, degrees/hour
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* *
*
||
||
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0
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4
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time, hour
||
+
+
100
Equivalent
statements
200
300
400
500
100
200
300
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500
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
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Example 10.1
Calculating the Center of Gravity
Rate of Change
Rate of temperature change, degrees/hour
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time, hour
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• Finding the center of gravity of a
structure is important in a number
of engineering applications
• The location of the center of
gravity can be calculated by
dividing the system up into small
components.
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
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xW  x1W1  x2W2  x3W3  etc...
yW  y1W1  y2W2  y3W3  etc...
Rate of Change
Rate of temperature change, degrees/hour
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time, hour
zW  z1W1  z2W2  z3W3  etc...
• In a rectangular coordinate system
• x , y , and z are the coordinates of the center of gravity
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500
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500
• W is the total mass of the system
• x1, x2, and x3 etc are the x coordinates of each system
component
•
y1, y2, and y3 etc are the y coordinates of each system
component
•
z1, z2, and z3 etc are the z coordinates of each system
component and
•
W1, W2, and W3 etc are the weights of each system
component
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
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In this example…
Rate of Change
Rate of temperature change, degrees/hour
-5
-4.5
-4
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-3
-2.5
-2
-1.5
-1
-0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time, hour
100
• We’ll find the center of gravity of a
small collection of the
components used in a complex
space vehicle
200
300
400
500
100
200
300
400
500
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
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Vehicle Component
Locations and Mass
Rate of Change
Rate of temperature change, degrees/hour
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time, hour
Item
x, meters y, meters
z meters
Mass
Bolt
0.1
2
3
3.50 gram
screw
1
1
1
1.50 gram
nut
1.5
0.2
0.5
0.79 gram
bracket
2
2
4
1.75 gram
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200
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500
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200
300
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500
Formulate the problem using a dot product
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
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transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
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Rate of Change
Rate of temperature change, degrees/hour
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time, hour
100
200
300
400
500
100
200
300
400
500
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
Rate of Change
Rate of temperature change, degrees/hour
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time, hour
100
200
300
400
500
100
200
300
400
500
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
We could use a plot to
evaluate our results
Rate of Change
Rate of temperature change, degrees/hour
-5
-4.5
-4
-3.5
-3
Center of Gravity
-2.5
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-1.5
-1
-0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
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5
time, hour
4
z-axis
3
This plot
was
enhanced
using the
interactive
plotting
tools
Center of Gravity
2
100
1
200
300
0
2
400
2
500
100
200
300
400
500
1
y-axis
1
0
0
x-axis
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
Matrix Multiplication
Rate of Change
Rate of temperature change, degrees/hour
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3
3.5
4
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5
time, hour
• Similar to a dot product
100
200
300
400
500
100
200
300
400
500
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
Rate of Change
Rate of temperature change, degrees/hour
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time, hour
100
200
300
400
500
100
200
300
400
500
• Matrix multiplication results in an
array where each element is a dot
product.
• In general, the results are found
by taking the dot product of each
row in matrix A with each column
in Matrix B
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
Rate of Change
Rate of temperature change, degrees/hour
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time, hour
N
Ci , j   Ai ,k Bk , j
k 1
100
200
300
400
500
100
200
300
400
500
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
Rate of Change
Rate of temperature change, degrees/hour
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time, hour
100
200
300
400
500
100
200
300
400
500
We could use matrix
multiplication to solve the
problem in Example 10.1, in
a single step
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
Matrix Powers
Rate of Change
Rate of temperature change, degrees/hour
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time, hour
• Raising a matrix to a power is
equivalent to multiplying it times itself
the requisite number of times
• A2 is the same as A*A
• A3 is the same as A*A*A
100
200
300
400
500
100
200
300
400
500
• Raising a matrix to a power requires it to
have the name number of rows and
columns
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
Matrix Inverse
Rate of Change
Rate of temperature change, degrees/hour
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time, hour
• MATLAB offers two approaches
• The matrix inverse function
• inv(A)
• Raising a matrix to the -1 power
100
• A-1
200
300
400
500
100
200
300
400
500
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
Rate of Change
Rate of temperature change, degrees/hour
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time, hour
Equivalent
approaches to
finding the
inverse of a
matrix
100
200
300
400
500
100
200
300
400
500
A matrix times its
inverse is the
identity matrix
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
Not all matrices have an
inverse
Rate of Change
Rate of temperature change, degrees/hour
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-3
-2.5
-2
-1.5
-1
-0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time, hour
• Called
• Singular
• Ill-conditioned matrices
100
200
300
400
500
100
200
300
400
500
• Attempting to take the inverse of a
singular matrix results in an error
statement
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
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Determinants
Rate of Change
Rate of temperature change, degrees/hour
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time, hour
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• Related to the matrix inverse
• If the determinant is equal to 0,
the matrix does not have an
inverse
• The MATLAB function to find a
determinant is
• det(A)
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
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Rate of Change
Rate of temperature change, degrees/hour
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time, hour
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500
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500
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
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Cross Products
Rate of Change
Rate of temperature change, degrees/hour
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• sometimes called vector products
• the result of a cross product is a
vector
• always at right angles (normal) to
the plane defined by the two input
vectors
• orthogonality
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
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Consider two vectors
Rate of Change
Rate of temperature change, degrees/hour
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


A  Ax i  Ay j  Az k



B  Bx i  B y j  Bz k
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300
The cross product is equal to…
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500
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


A  B  ( Ay * Bz  Az * B y )i  ( Az * Bx  Ax * Bz ) j  ( Ax B y  Ay Bx )k
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
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transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
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Rate of Change
Rate of temperature change, degrees/hour
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time, hour
100
200
300
400
500
100
200
300
400
500
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
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Cross Products are Widely
Used
Rate of Change
Rate of temperature change, degrees/hour
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0
0
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4
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time, hour
100
• Cross products find wide use in
statics, dynamics, fluid mechanics
and electrical engineering
problems
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500
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MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
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10.2 Solutions to Systems of
Linear Equations
Rate of Change
Rate of temperature change, degrees/hour
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3x 2 y
x
x
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200
 z  10
3y 2 z  5
 y  z  1
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500
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MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
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Using Matrix Nomenclature
Rate of Change
Rate of temperature change, degrees/hour
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 3 2  1
 x
A   1 3 2  X   y 
 z 
 1  1  1
10 
B   5 
 1
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and
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500
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AX=B
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
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We can solve this problem using the
matrix inverse approach
Rate of Change
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This approach is easy
to understand, but its
not the more efficient
computationally
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
Rate of Change
Rate of temperature change, degrees/hour
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0
0.5
1
1.5
2
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3
3.5
4
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5
time, hour
100
Matrix left division
uses Gaussian
elimination, which
is much more
efficient, and less
prone to round-off
error
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500
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
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10.3 Special Matrices
Rate of Change
Rate of temperature change, degrees/hour
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time, hour
• We introduced some of
MATLAB’s special matrices in
previous chapters
• ones
• zeros
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MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
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The identity matrix is another
special matrix that is useful in Matrix
Algebra
Rate of Change
Rate of temperature change, degrees/hour
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It may be tempting
to name an identity
matrix i, however i
is already in-use
for imaginary
numbers
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MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions
Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.