1. Solving Simultaneous Equations by Matrix Inverse Problem 2.5 # 37 Presented by E. G. Gascon

2. The Problem

3. How to solve 246 -4-3 01 To solve a system of equations AX = B, where A is the matrix of coefficients. B is the matrix of constants. 4 8 -4 Need to find A -1 the inverse of A So… solving for X = A -1 B

4. Finding A -1 246100 -4-3010 01001 Start by setting up the matrix with the identity in the last three columns 1001.752.53 010-0.25-0.50 001-0.25-0.5 After performing Gauss-Jordan, the resulting matrix is A -1 See next slide for Gauss- Jordan Work

5. Gauss –Jordan to find A -1 246100 -4-3010 01001 246100 r1 +2r2 0-40120 01001 r2+r1 206220 0-40120 r2+4r3 00-4124 3r3 + 2r1 40071012 0-40120 00 124 r1 / 4 1001.752.53 r2 / -4 010-0.25-0.50 r3 / -4 001-0.25-0.5

6. Alternate way to get A -1 using Excel Function MINVERSE 246 -4-3 01 ___ ___ -___ Step 1: Enter the 3X3 coefficient matrix in an Excel Spread Sheet. Step 2: Select a 3X3 matrix of empty cells and type =MINVERSE( matrix is step 1) Ctrl + Shift + ENTER 1.752.53 -0.25-0.50 -0.25-0.5

7. Solve for X matrix X = A -1 B 1.752.53 -0.25-0.50 -0.25-0.5 4 8 -4 Times 15 -5 = Ans: (15, -5, -1) Use matrix multiplication or MMULT function in Excel.