Fundamentals of Engineering Analysis

1. Fundamentals of Engineering Analysis EGR 1302 - Adjoint Matrix and Inverse Solutions, Cramer’s Rule

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. Cramer’s Rule In many instances of complex problems, we may only need a partial solution. As we have seen, calculating an inverse takes a lot of computing power. However, calculating the determinant is much more manageable. Before the days of electronic computers, mathematician Gabriel Cramer devise a shortcut to the solution of linear systems. It also gives an explicit expression for the solution of the system Gabriel Cramer (1704-1752).

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

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

10. the system of equations in matrix form is - Solve a System of Equations with Cramer’s Rule Remember: “ratio of determinants”

11. Questions?