© 2005 Baylor University Slide 1 Fundamentals of Engineering Analysis EGR Adjoint Matrix and Inverse Solutions, Cramer’s Rule

© 2005 Baylor University Slide 2 The Adjoint Matrix and the Inverse Matrix Recall the Rules for the Inverse of a 2x2: 1.Swap Main Diagonal 2.Change sign of a 12, a 21 3.Divide by determinant 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

© 2005 Baylor University Slide 3 Calculating the Adjoint Matrix and A detA Problem 7.13 in the Text adjA =

© 2005 Baylor University Slide 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 = x2s. (Wow!) Which is why we use computers (and explains why so many problems could not be solved before the advent of computers).

© 2005 Baylor University Slide 5 Class Exercise: Find the Adjoint of A Work this out yourself before going to the solution on the next slide

© 2005 Baylor University Slide 6 Class Exercise: Solution Notice that: detA = 0, therefore matrix A is singular. However, even though the Determinant is zero, the Adjoint still exists. This means that the Inverse does not exist.

© 2005 Baylor University Slide 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 ( ).

© 2005 Baylor University Slide 8 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 9 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 10 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 11 Questions?