Programming Part: For each programming problem, please submit the code and the output by pasting them into a word document and attaching them to your homework.

1. Implement the back-substitution algorithm in Matlab to solve Ux = b where U = [uij]1010, and uij =(cos(ij), i ? j 0, i > j , b = [bi1]101,bi1 = tan(i).

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2. Gauss-Jordan Method: This method is used to solve linear systems as an alternative to Gaussian Elimination. In addition to using pivot element a(k?1) kk to eliminate every element below in the kth column, but also use it to eliminate every element above it. This results in the augmented matrix being diagonal and therefore back-substitution is not needed (instead xk = b(n?1) k a(n?1) kk for k = 1,...,n) ? ? ? ? ? ? ? a(n?1) 11 0 ... 0 b(n?1) 1 0 a(n?1) 22 ... . . . b(n?1) 2 . . . ... ... 0 . . . 0 ... 0 a(n?1) nn b(n?1) n ? ? ? ? ? ? ? Modify the Gaussian Elimination code to incorporate the Gauss-Jordan Method and re-solve problem 2 (above) to check your work.

Programming Part: For each programming problem, please submit the code and the output by pasting them into a word document and attaching them to your homework. 1. Implement the backsubstitution algorithm in Matlab to solve U): = b where ija 2. GaussJordon Method: This method is used to solve Jjnear systems as an alternative to Gaussian Elimination. In addition to using pivot element Gistl} to eliminate everg,r element below in the lath oolunm, but also use it to eliminate every element above it. This results in the augmented matrix being Ln-IJ diagonal and therefore backsubstitution is not needed (instead :3,: = b n_ for k. = 11 ...,n} 'kk egg'3' n ... 0 59"\" 0 egg") E 53\"\" . ' . . 0 . {I .-. D 15,1\" bill-1:] Modify the Gaussian Elimination code to inoorporate the GaussJordan Method and resolve problem 2 (above) to check your work