Fundamentals of Engineering Analysis EGR 1302 Unit 1, Lecture D

1. Fundamentals of Engineering Analysis EGR 1302 Unit 1, Lecture D Approximate Running Time - 22 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. 3 positive terms 3 negative terms Determinants - A Property of a Square Matrix “Eyeball” Method

3. Determinant of a 3x3 Let’s factor out the elements of the first row of the matrix, i.e.

4. We can identify this construct as the “Cofactor” Determinant of a 3x3

5. Every element in a square matrix has a cofactor The Cofactor Matrix of a 3x3 The cofactor of any element is “the determinant formed by striking out the Row & Column of that element

6. Sign of the Cofactor: The Cofactor Matrix of a 3x3 Caution: Do not forget the signs of the cofactors

7. Determinant by Row Expansion using the first row: Row Expansion:

8. Using the TI-89 to find Determinants We had previously entered a matrix and assigned it to the variable “a” The calculator has the built-in function “det()“ Which calculates the determinant of a square matrix.

9. Determinant by Row or Column Expansion Select Any Row or Column to do the Expansion Pick Column #1 to simplify the calculation due to the zero terms.

10. Finding the Cofactor Matrix of A Calculators and Computers obviously make this process easier. 

11. Similar, but not quite Rules for 2x2 Inverse and the Cofactor Matrix 1. Swap Main Diagonal 2. Change Signs on a12, a21 3. Divide by detA

12. Properties of Determinants 1. Determinant of the Transpose Matrix det A = det AT

13. for Properties of Determinants 2. Multiply a single Row (Column) by a Scalar - k det B = k*det A det B = 3*det A

14. swap 4. Expansion by any Rows (Columns) equals the same Determinant Properties of Determinants 3. If two Rows (Columns) are swapped, the sign changes det B = -det A Recall:

15. det B = 0 det A = 0 Col2 = 2*Col1 Row2 = Row1 Properties of Determinants 5. If two Rows (Columns) are equal, or the same ratio, i.e., Row1 = k*Row2 det A = 0 The matrix A is “singular” But if detA=0, a unique solution does not exist Recall Rule #3 to find A-1, divide by detA

16. Construct D by creating a new Row 2 Properties of Determinants • If a new matrix B is constructed from A • by adding K*rowi to another rowj … det B = det A These are called Row (Column) Operations

17. Finding the Determinant: Two Methods “Eyeball” Method 2 + (-40) + (-6) – (-5) -12 –(-8) = -43 Row Expansion 1*(2-12) -2(-4+3) -5(8-1) -10 + 2 -35 = -43

18. This concludes Unit 1, Lecture D