Fundamentals of Engineering Analysis EGR 1302 - Adjoint Matrix and Inverse Solutions

1. Fundamentals of Engineering Analysis EGR 1302 - Adjoint Matrix and Inverse Solutions Approximate Running Time - 7 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. If the Cofactor Matrix is “transposed”, we get the same matrix as the Inverse And we define the “Adjoint” as the “Transposed Matrix of Cofactors”. And we see that the Inverse is defined as The Adjoint Matrix and the Inverse Matrix • Recall the Rules for the Inverse of a 2x2: • Swap Main Diagonal • Change sign of a12, a21 • Divide by determinant

3. Problem 7.13 in the Text Calculating the Adjoint Matrix and A-1 adjA = -12 detA

4. Complexity of Large Matrices Consider the 5x5 matrix, S • To find the Adjoint of S (in order to find the inverse), would require • Finding the determinants of 25 4x4s, which means • Finding the determinants of 25*16 = 400 3x3s, which means • Finding the determinants of 400*9 = 3600 2x2s. (Wow!) Which is why we use computers (and explains why so many problems could not be solved before the advent of computers).

5. Class Exercise: Find the Adjoint of A Work this out yourself before going to the solution on the next slide

6. This means that the Inverse does not exist. Class Exercise: Solution Notice that: detA = 0, therefore matrix A is singular. However, even though the Determinant is zero, the Adjoint still exists.

7. This concludes the Lecture