© 2005 Baylor University Slide 1 Fundamentals of Engineering Analysis EGR Determinants Approximate Running Time - 22 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 Determinants “Eyeball” Method 3 positive terms 3 negative terms - A Property of a Square Matrix

© 2005 Baylor University Slide 3 Determinant of a 3x3 Let’s factor out the elements of the first row of the matrix, i.e.

© 2005 Baylor University Slide 4 Determinant of a 3x3 We can identify this construct as the “Cofactor”

© 2005 Baylor University Slide 5 The Cofactor Matrix of a 3x3 The cofactor of any element is “the determinant formed by striking out the Row & Column of that element Every element in a square matrix has a cofactor

© 2005 Baylor University Slide 6 The Cofactor Matrix of a 3x3 Sign of the Cofactor: Caution: Do not forget the signs of the cofactors

© 2005 Baylor University Slide 7 Determinant by Row Expansion Row Expansion: using the first row:

© 2005 Baylor University Slide 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.

© 2005 Baylor University Slide 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.

© 2005 Baylor University Slide 10 Finding the Cofactor Matrix of A Calculators and Computers obviously make this process easier.

© 2005 Baylor University Slide 11 Rules for 2x2 Inverse and the Cofactor Matrix 1. Swap Main Diagonal 2. Change Signs on a 12, a Divide by detA Similar, but not quite

© 2005 Baylor University Slide 12 Properties of Determinants 1. Determinant of the Transpose Matrix det A = det A T

© 2005 Baylor University Slide 13 Properties of Determinants 2. Multiply a single Row (Column) by a Scalar - k det B = k*det A for det B = 3*det A

© 2005 Baylor University Slide 14 Properties of Determinants 3. If two Rows (Columns) are swapped, the sign changes det B = -det A swap Recall: 4. Expansion by any Rows (Columns) equals the same Determinant

© 2005 Baylor University Slide 15 Properties of Determinants 5. If two Rows (Columns) are equal, or the same ratio, i.e., Row 1 = k*Row 2 det A = 0 det B = 0 Col 2 = 2*Col 1 det A = 0 Row 2 = Row 1 The matrix A is “singular” Recall Rule #3 to find A -1, divide by detA But if detA=0, a unique solution does not exist

© 2005 Baylor University Slide 16 Properties of Determinants 6.If a new matrix B is constructed from A by adding K*row j to another row i … det B = det A Construct D by creating a new Row 2 These are called Row (Column) Operations

© 2005 Baylor University Slide 17 Finding the Determinant: Two Methods 2 + (-40) + (-6) – (-5) -12 –(-8) = -43 “Eyeball” Method Row Expansion 1*(2-12) -2(-4+3) -5(8-1) = -43

© 2005 Baylor University Slide 18 This concludes the Lecture