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Fundamentals of Engineering Analysis EGR 1302 Unit 1, Lecture B

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:

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Fundamentals of Engineering Analysis EGR 1302 Unit 1, Lecture B

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  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: • Select “Slide Show” with the menu: Slide Show|View Show (F5 key), and hit “Enter” • 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” • 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. A= B= For A: m=2, n=3 For B: n=3, p=4 2 x 3 3 x 4 For A * B, n=n; i.e. 3=3, so A*B is “conformable” Note that B * A is Undefined (not allowed) because p = m 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

  3. Because matrices must be conformable for multiplication; in general A * B = B * A 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)

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

  5. Matrix Multiplication * = 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 Remember:

  6. Matrix Multiplication A * B = 2 x 2 B * A = 3 x 3

  7. Matrix Multiplication A * B = 9 5 * = 1 7

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

  9. Sum of Products form Linear Systems as Sum of Products ax1 + bx2 + cx3 = d x1 x2 x3 [ a b c ] - a 1 x 3 row vector - a 3 x 1 column vector * x1 x2 x3 [ a b c ] = [ d ] - a 1 x 1 scalar – i.e.; ax1 + bx2 + cx3 = d

  10. Conformability and Order of Matrix Multiplication B4x5 A5x4 C6x4 Given: 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

  11. * = 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] * Properties of a Zero Matrix = Note that:

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

  13. Special Matrices The Transpose Matrix Rule: The Row becomes the Column, and the Column becomes the Row A is a 2x3, so AT will be a 3x2 For a 3x3

  14. Properties of the Transpose Matrix A*B= AT*BT = ? BT*AT = (AB)T= BT*AT

  15. Additional Properties of the Transpose If A+B and A*B are allowed (are conformable), then (A+B)T = AT + BT (AB)T = BTAT

  16. The Diagonal The Symmetric Matrix Must be Square: n x n A = AT A + AT must also be Symmetric

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

  18. The Identity Matrix Must be Square: n x n And must be Diagonal Notation: IN Can be any Order The Unity term A*I = A I*A = A A does not have to be square Amxn * In = A or Im * Amxn = A

  19. = * Powers of Matrices A * A = A2 for Square Matrices Only A * A2 = A3 … and so on If A is Diagonal … A2 = a112, a222, a332

  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. Matrix Math on the TI-89 Calculator B*A = ? A*B – not conformable

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

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

  24. Using the Matrix Editor on the TI-89

  25. This concludes Unit 1, Lecture B

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