1. © 2005 Baylor University Slide 1 Fundamentals of Engineering Analysis EGR 1302 Unit 1, Lecture B Approximate Running Time - 24 minutes Distance Learning / Online Instructional Presentation Presented by Department of Mechanical Engineering Baylor University Procedures: 1.Select “Slide Show” with the menu: Slide Show|View Show (F5 key), and hit “E nter” 2.You will hear “CHIMES” at the completion of the audio portion of each slide; hit the “Enter” key, or the “Page Down” key, or “Left Click” 3.You may exit the slide show at any time with the “Esc” key; and you may select and replay any slide, by navigating with the “Page Up/Down” keys, and then hitting “Shift+F5”.

2. © 2005 Baylor University Slide 2 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 B Which defines the order of the multiplication A= 2 x 3 For A: m=2, n=3B= 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”

3. © 2005 Baylor University Slide 3 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

4. © 2005 Baylor University Slide 4 Matrix Multiplication m x n 2 x 3 m x p n x p 3 x 3 = A * B = C is Conformable Is a Row on Column operation The Product C will be a 2 x 3

5. © 2005 Baylor University Slide 5 Matrix Multiplication * = C 11 is made up of Row 1 from A, and Column 1 from B Note the “sum of products” form C 12 is made up of Row 1 from A, and Column 2 from B Remember:

6. © 2005 Baylor University Slide 6 Matrix Multiplication A * B = B * A = 3 x 3 2 x 2

7. © 2005 Baylor University Slide 7 * = 9 5 1 7 Matrix Multiplication A * B =

8. © 2005 Baylor University Slide 8 Matrix Multiplication 4 3 2 1 1 4 11 B * A = * = Work this out yourself, before proceeding, To make sure you understand the method of matrix multiplication.

9. © 2005 Baylor University Slide 9 Linear Systems as Sum of Products ax 1 + bx 2 + cx 3 = d Sum of Products form [ a b c ] - a 1 x 3 row vector x1x2x3x1x2x3 - a 3 x 1 column vector [ a b c ] * = [ d ]- a 1 x 1 scalar – i.e.; x1x2x3x1x2x3 ax 1 + bx 2 + cx 3 = d

10. © 2005 Baylor University Slide 10 Conformability and Order of Matrix Multiplication Given: A 5x4 B 4x5 C 6x4 A * B = D 5x5 B * A = E 4x4 A * C = not conformable C * A = not conformable C * B = F 6x5 A * B * C = not conformable C * B * A = G 6x4

11. © 2005 Baylor University Slide 11 Properties of a Zero Matrix * = 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: * =

12. © 2005 Baylor University Slide 12 Matrix Form of Linear Equations Distributive Property: A(B+C) = AB + AC Associative Property: A(BC) = (AB)C Thencan become A * = Any Order ? How do we solve this system of equations

13. © 2005 Baylor University Slide 13 Special Matrices The Transpose Matrix Rule: The Row becomes the Column, and the Column becomes the Row A is a 2x3, so A T will be a 3x2 For a 3x3

14. © 2005 Baylor University Slide 14 Properties of the Transpose Matrix A*B= A T *B T = ? B T *A T = (AB) T = B T *A T

15. © 2005 Baylor University Slide 15 Additional Properties of the Transpose If A+B and A*B are allowed (are conformable), then (A+B) T =A T + B T (AB) T = B T A T

16. © 2005 Baylor University Slide 16 The Symmetric Matrix A = A T Must be Square: n x n A + A T must also be Symmetric The Diagonal

17. © 2005 Baylor University Slide 17 The Diagonal Matrix Must be Square: n x n All off-diagonal elements Are Zero If A and B are Diagonal + A+B will be Diagonal = If A and B are Diagonal * A*B will be Diagonal =

18. © 2005 Baylor University Slide 18 The Identity Matrix Must be Square: n x n And must be Diagonal Can be any Order Notation: I N The Unity term A*I = A I*A = A A does not have to be square A mxn * I n = A or I m * A mxn = A

19. © 2005 Baylor University Slide 19 Powers of Matrices A * A = A 2 for Square Matrices Only A * A 2 = A 3 … and so on If A is Diagonal … A 2 = a 11 2, a 22 2, a 33 2 = *

20. © 2005 Baylor University Slide 20 Matrix Math on the TI-89 Calculator My Philosophy for using Calculators (and Computers …) Be aware of the Order of Magnitude Sign Errors are easy to miss Double check your work If you understand the solution methodology, You will understand the answer.

21. © 2005 Baylor University Slide 21 A*B – not conformable B*A = ? Matrix Math on the TI-89 Calculator

22. © 2005 Baylor University Slide 22 Matrix Math on the TI-89 Calculator (cont.)

23. © 2005 Baylor University Slide 23 Matrix Math on the TI-89 Calculator (cont.)

24. © 2005 Baylor University Slide 24 Using the Matrix Editor on the TI-89

25. © 2005 Baylor University Slide 25 This concludes Unit 1, Lecture B