Sage Project 3: for the third project, please complete the following tasks:

- Watch the Project 3 Sage video.
- Find three different echelon forms for the matrix $\begin{pmatrix}2& 1 &7 &-7& 2\\ -3& 4& -5 &-6 & 3\\ 1& 1& 4 &-5 &2\end{pmatrix}$.
- Use the process of Gaussian Elimination (showing each step) to find the reduced echelon form for the matrix $\begin{pmatrix}1 &0 &-3&0&6&0&7&-5& 9\\ 0&0& 0& 5&0&1&0& 3& -7\\ 0&0& 0& 0&0&0&0& 0& 0\\ 0&1& 0& 0&0&0&0&-4 &2\\0& 0& 0& 0& 0& 0& 1& 7& 3\\ 0&0&0&0&0&0&0&0&0\end{pmatrix}$.
- Check that the previous answer is correct by using the built in functionality of Sage to compute the echelon form (note: echelon_form() does not necessarily give the reduced echelon form; instead use .rref() appended to a matrix for reduced row echelon form.)

- Consider the system of (three) linear equations: $2x_1 +x_2 +7x_3 -7x_4=8;\ \ -3x_1 +4x_2 -5x_3 -6x_4=−12;\ \ x_1 +x_2 +4x_3 -5x_4=4$
- Use the built in "solve" function to solve this system.
- Use the echelon form of an appropriate matrix to solve this system.
- Are the answers the same? Which method is easier to use? Why?
- Write the general solution to the system in standard form.

- For the matrix $\begin{pmatrix}2&3&5\\ -1& 4& 6\\ 3& 10& 2\\ 3& -1 &-1\\ 6&9&3\end{pmatrix}$,
- Find the rank using the echelon form.
- Find the rank using the built in function.
- Suppose the matrix represents an augmented matrix for a system of linear equations. Is there a solution to this system? Explain.

- Choose four different 4 by 4 matrices and find their reduced echelon forms. Did anything interesting happen?
- Read Chapter 1.6.1 on curve fitting in the book. Then complete problem 19 from 1.6 using Sage.
- Find two 2 by 2 matrices, M and N, such that M*N is not the same as N*M.
- Find two matrices, A and B, of any size so that you can compute A*B but you cannot compute B*A. Check your work with Sage.