Week 4, Day 2

Today we learned how to form augmented matrices and solve for their variables through two methods: Gaussian Elimination with Back Substitution and Gauss-Jordan elimination. When performimg Gaussian Elimination, you should consider the following steps: First, write the augmented matrix of the system of linear equations. Second, use elementary row operations to rewrite the augmented matrix in row-echelon form. Finally, write the system of linear equations corresponding to the matrix in row-echelon form, and use back-substitution to find the solution. Ultimately, this method requires that the main diagonal of the matrix consists of one's and everything below it are zeros. The Gauss-Jordan Elimination method, however, requires that the main diagonal of the matrix consists of one's and everything around it are zeros. The steps to performing Gauss-Jordon are similar to that of Gaussian Elimination. You simply continue using elementary row operations to achieve the appropriate form until each variable has been solved for without the use of back substitution. 

In the above problem, the student didn't quite understand the process of Gauss-Jordan Elimination. While he has achieved row-echelon form, the student forgot that the operations he used apply to the solutions on the right side of the dotted line as well. When he multiplies -2 by row one and adds it to row 2, he must remember to do so for not only 2 and 3 in the second row, but also the  5. The operation would yield a solution of -8 + 5 = 3. The same goes for the first row in which he multiplies row 2 by -1 and adds it to row 1. This operation would yield a solution of -5 + 4 = -1. The final solutions would be -1 for the first variable and 3 for the second variable. 

Problem 1:

Problem 2:

Problem 3:

I used the Gauss-Jordan Elimination method in problems 1 and 2, and the Gaussian Elimination method in problem 3.