Use the reduced row echelon form $($RREF$)$ to solve Ax $=$ b$,$ where A and b are indicated in Exercise $1,$ and type your answer for the following in the Live Editor.

Display the reduced row echelon form and the pivot columns of the augmented matrix $[$A b$].$

Write a report to explain if there is a solution of Ax $=$ b based on rref$([$A b$]).$

Verify the Rouch$-$Capelli Theorem by comparing the rank$($A$)$ and rank$([$A b$]).$

Call the function rank$\_$comp and determine if Ax $=$ b is consistent.

Compare the result with Part A$.$ You should have the same conclusion.

Note: rank$\_$comp gives you a new command to compare if rank$($A$)=$ rank$([$A b$]).$ When you need to use the command, read the comments in the m$-$file function first and call the function by typing the name of the function, rank$\_$comp$($A$,$B$),$ in the Live Editor where A and B are the two input matrices.

Open the template LS$\_$solution. Use the Rouch$-$Capelli Theorem as a guide and code the function using an if$-$elseif statement $($do not use nested if$-$else statements$).$ When LS$\_$solution is called it should return the correct system type

In this exercise, you use the RREF and Rouch$-$Capelli Theorem to determine if Ax $=$ b is consistent. You may need the following commands for this exercise.

The command rref$($A$)$ returns the reduced row echelon form of a matrix A

The command $[$B$,$ pivcols$]=$rref$($A$)$returns B and pivcols, where matrix B $=$ rref$($A$)$ and pivcols indicates pivot columns of A

The command rank$($A$)$ returns the rank of a matrix A

You also need two m$-$file functions rank$\_$comp $($given$)$ and LS$\_$solution $($template only$).$ In these functions, you need to know:

if$-$else statement

If an expression is true, it executes statements $1$ block. Otherwise, it executes statements $2$ block.

if expression

statements $1$ block

else

statements $2$ block

end

if$-$elseif statement

If expression $1$ is true, it executes statements $1$ block. If not, check expression $2.$ If expression $2$ is true, it executes statements $2$ block. If not, execute statements $3$ block.

if expression $1$

statements $1$ block

elseif expression $2$

statements $2$ block

else

statements $3$ block

end

Use the reduced row echelon form $($RREF$)$ to solve Ax $=$ b$,$ where A and b are indicated in Exercise $1,$ and type your answer for the following in the Live Editor.

Display the reduced row echelon form and the pivot columns of the augmented matrix $[$A b$].$

Write a report to explain if there is a solution of Ax $=$ b based on rref$([$A b$]).$

Verify the Rouch$-$Capelli Theorem by comparing the rank$($A$)$ and rank$([$A b$]).$

Call the function rank$\_$comp and determine if Ax $=$ b is consistent.

Compare the result with Part A$.$ You should have the same conclusion.

Note: rank$\_$comp gives you a new command to compare if rank$($A$)=$ rank$([$A b$]).$ When you need to use the command, read the comments in the m$-$file function first and call the function by typing the name of the function, rank$\_$comp$($A$,$B$),$ in the Live Editor where A and B are the two input matrices.

Open the template LS$\_$solution. Use the Rouch$-$Capelli Theorem as a guide and code the function using an if$-$elseif statement $($do not use nested if$-$else statements$).$ When LS$\_$solution is called it should return the correct system type based on whether Ax $=$ b has a solution and how many. The function should output$/$return$/$set system$\_$type to:

inc if the system is inconsistent and has no solution, or

con$\_$with$\_$one$\_$sol if the system is consistent and has a unique solution, or

con$\_$with$\_$inf$\_$sols if the system is consistent and has infinitely many solutions.

Note: You should include three inputs $($n$,$ A$,$ Ab$)$ for LS$\_$solution, where n is the number of variables in the system of equation.