# Objective: Write code that puts a matrix into reduced-row echelon f...

## Transcribed Text

Objective: Write code that puts a matrix into reduced-row echelon form. It should work for all sizes of matrices. Criteria: There will be loops, of course. Nowhere should you use MATLAB's built in row reduction functions. Your code needs to include a check to insure that you are not dividing by zero. It needn't be able to correct this issue on the fly. Problems: Use your code to solve the following systems of equations. x-2y+3z=9 -x+3y = - 5y + 5z = 17 x +3y=2 -x+2y=3 X2 +x-2x4=-3 - X1 + 2x2 - X3 = 2 + 4x2 + X3 - 3x4 = X1-4x2-7x3-X4=-19 - Objective: Write code to put an augmented matrix into row-echelon form (more on that below). Criteria: Must be a function, with loops, that takes an augmented matrix as it's input and outputs the row-echelon form of this matrix. Objective: Write code to solve a matrix in row-echelon form using backwards substitution. Critera: Must be a function that takes a row-echelon form matrix and output the solution to the original system of equations. An augmented matrix is in row-echelon form if it looks like this [11 @12 @13 b1 0 @22 @23 b2 a33 b3. The important feature is that the lower triangle is all zeroes. With the matrix in this form, it corresponds to the following system of equations. @11X1 + a12X2 + a13X3 = b1 a22X2 + a23X3 = b2 a33X3 = b3 It should be readily apparent that this system can be solved rapidly in this form. The bottom equation has the solution X3 = by Now X3 can be plugged into the second equation to find the solutions b3 @33 _be-costan) = and then b1 - @13 @12 @22 X3 = @11 This method is called backwards substitution. It involves finding one solution and then using it to find the next, and so on.. FINAL Objective: Compare your two system solving algorithms. You have one from last week that puts augmented matrices into row-reduced echelon form and now this one which uses row echelon form plus backwards substitution. Which do you think is more efficient? Find out for sure using the tic/toc commands built into MATLAB. Criteria: Use the tic and toc commands to time your algorithms. Practice using calling tic, then toc in the command window to get a feel for how it works. Place the tic command at the beginning of your primary solving function and then toc right at the very end. This will ouput and overall algorithm execution time. Do this for both system solving methods to compare their efficiencies. Problem: Compare the run-times of both methods for different sizes of systems. Which is faster? By how much? Does the size of the system have an effect on this? In what way?

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clear; clc; close all;
%Solutions
%%
%Solution to Linear System 1
A=[1 -2 3;-1 3 0;2 -5 5]; B=[9 -4 17]';
A_aug=[A B];
Af=gauself(A_aug);%Do gaussian forward elimination to get row reduced Echelon form
fprintf('\nAugmented matrix of the given system is as follows.\n');
A_aug
fprintf('\nRow Echelon form of the given system is as follows.\n');
Af
[Ab,X]=gausel(Af);%Do back substitution of the Row Echelon form matric obtained above and solve system
fprintf('\nSolution to given system is as follows.\n');
X
%%
%Solution to Linear System 2
A=[1 3;-1 2]; B=[2 3]';
A_aug=[A B];
Af=gauself(A_aug);%Do gaussian forward elimination to get row reduced Echelon form
fprintf('\nAugmented...

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