© 2005 Baylor University Slide 1 Fundamentals of Engineering Analysis EGR 1302 Unit 1, Lecture F 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 “Enter” 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”.

© 2005 Baylor University Slide 2 solved as = Solving Systems of Linear Equations given a system becomes by row expansion etc. which is the same form as: for, replace in Col 1. where

© 2005 Baylor University Slide 3 Solution by Cramer’s Rule Replace Col. 3 for Replace Col. 1 for Replace Col. 2 for Cramer’s Rule is only valid for Unique Solutions. If detA = 0, Cramer’s Rule fails! Cramer’s Rule: Replace in the column # of the unknown variable you wish to find and solve for the “Ratio of Determinants”.

© 2005 Baylor University Slide 4 Solve a System of Equations with Cramer’s Rule the system of equations in matrix form is - Remember: “ratio of determinants”

© 2005 Baylor University Slide 5 Cramer’s Rule is only valid for Unique Solutions. If detA = 0, Cramer’s Rule fails! The Need for a General Solution to Linear Systems Unique Solution, all cross at the same point, the “solution” Det = 0 We need a method of finding a general solution when the coefficient matrix A is Singular.

© 2005 Baylor University Slide 6 Gaussian Elimination - A general solution Methodology Elementary Row Operations: 1. Multiply by a constant - 2. Swap two rows - 3. Replace a row by adding to it another k*row - Eqn from the Text We will use three elementary row operations to solve this set of linear equations by Gaussian Elimination.

© 2005 Baylor University Slide 7 Using Elementary Row Operations to Solve by Gaussian Elimination Step 1: Use Rule 3 to eliminate from rows 2 & 3: 1. Keep Row 1 the same Step 2: Use Rule 3 to eliminate from row 3: 1. Keep Row 1 the same Keep Row 2 the same Step 3: Use Rule 1 to reduce all coefficients to 1: 1. Keep Row 1 the same 3. 2.

© 2005 Baylor University Slide 8 The Augmented Matrix can be represented as “augmented” matrix Step Keep Row 1 the same Step Keep Row 1 the same Keep Row 2 the same Step Keep Row 1 the same “Row Echelon Form”

© 2005 Baylor University Slide 9 Reduced Row Echelon Form of the Augmented Matrix Using “Backwards Substitution” on the Row Echelon Form Identity Matrix “Reduced Row Echelon Form” Observe directly that

© 2005 Baylor University Slide 10 Using the TI-89 to do Gaussian Elimination Note that calculator computes a different REF result, by using a different algorithm, but the answer is still correct. Save the augmented matrix as variable “x34” Reduced Row Echelon Form - use the function “rref() Row Echelon Form - use the function “ref() TI-89 Result Manual Result

© 2005 Baylor University Slide 11 This concludes Unit 1, Lecture F