Problem 1

1. write down augmented matrix of th linear system $$\begin{array}{r}\{\begin{array}{ll}2x_1+x_3+21x_4& =5\\ x_1+9x_2+8x_3+3x_4& =6\\ 3x_1-12x_4& =0\end{array}\end{array}$$
   • as of following: $$\left[\begin{array}{ccccc}2& 0& 1& 21& 5\\ 1& 9& 8& 3& 6\\ 3& 0& 0& 12& 0\end{array}\right]$$
2. compute the reduced echelon form of the matrix

• to echelon form $$\begin{array}{rl}\left[\begin{array}{ccccc}2& 0& 1& 21& 5\\ 1& 9& 8& 3& 6\\ 3& 0& 0& 12& 0\end{array}\right]& =\left[\begin{array}{ccccc}3& 0& 0& 12& 0\\ 2& 0& 1& 21& 5\\ 1& 9& 8& 3& 6\end{array}\right]\\ & =\left[\begin{array}{ccccc}1& 0& 0& 4& 0\\ 2& 0& 1& 21& 5\\ 1& 9& 8& 3& 6\end{array}\right]\\ & =\left[\begin{array}{ccccc}1& 0& 0& 4& 0\\ 2& 0& 1& 21& 5\\ 0& 9& 8& -1& 6\end{array}\right]\\ & =\left[\begin{array}{ccccc}1& 0& 0& 4& 0\\ 0& 0& 1& 13& 5\\ 0& 9& 8& -1& 6\end{array}\right]\\ & =\left[\begin{array}{ccccc}1& 0& 0& 4& 0\\ 0& 9& 8& -1& 6\\ 0& 0& 1& 13& 5\end{array}\right]\end{array}$$
• to RREF $$\begin{array}{rl}\left[\begin{array}{ccccc}1& 0& 0& 4& 0\\ 0& 9& 8& -1& 6\\ 0& 0& 1& 13& 5\end{array}\right]& =\left[\begin{array}{ccccc}1& 0& 0& 4& 0\\ 0& 9& 0& -105& -34\\ 0& 0& 1& 13& 5\end{array}\right]\\ & =\left[\begin{array}{ccccc}1& 0& 0& 4& 0\\ 0& 1& 0& -35/3& -34/9\\ 0& 0& 1& 13& 5\end{array}\right]\end{array}$$

3. how many solutions exists?
   • infinite, as there is a free variable, and the last column does not have a pivot

Problem 2

• find all solutions to following system $$\begin{array}{r}\{\begin{array}{ll}{x}_1-x_2-6x_3& =10\\ 2x_2+7x_3& =-10\\ x_1+x_2+x_3& =0\end{array}\end{array}$$
• to matrix $$\begin{array}{rl}\left[\begin{array}{cccc}1& -1& -6& 10\\ 0& 2& 7& -10\\ 1& 1& 1& 0\end{array}\right]& =\left[\begin{array}{cccc}1& -1& -6& 10\\ 0& 2& 7& -10\\ 0& 2& 7& -10\end{array}\right]\\ & =\left[\begin{array}{cccc}1& -1& -6& 10\\ 0& 2& 7& -10\\ 0& 0& 0& 0\end{array}\right]\\ & =\left[\begin{array}{cccc}1& -1& -6& 10\\ 0& 1& 7/2& -5\\ 0& 0& 0& 0\end{array}\right]\\ & =\left[\begin{array}{cccc}1& 0& -5/2& 5\\ 0& 1& 7/2& -5\\ 0& 0& 0& 0\end{array}\right]\end{array}$$
• infinitely many solution, as there is a free variable & last column is not a pivot