Consider the following augmented matrix
\[
M=\left(\begin{array}{ccc|c}
2 & 2 & 0 & 9 \\
2 & 2 & -2 & 11 \\
-2 & 2 & 2 & -9
\end{array}\right)
\]
(i) Write out the system corresponding to the given augmented matrix with variables ordered as \( (x, y, z) \).
(The system is written as a list enclosed in block brackets and the equations are separated by commas, i.e.
\[
\left.\left[a^{\star} x+b^{\star} y+c^{\star} z=d, f^{*} x+g^{\star} y+h^{\star} z=j, k^{\star} x+I^{*} y+m^{\star} z=n\right]\right)
\]
(ii) Suppose the following sequence of elementary row operations (ERO) are used to reduce the augmented matrix to echelon form;
\[
\begin{array}{l}
R_{2}-R_{1} \rightarrow R_{2}, R_{3}+R_{1} \rightarrow R_{3}, R_{2} \leftrightarrow R_{3} \\
\frac{1}{2} R_{2} \rightarrow R_{2}, \frac{1}{a} R_{1} \rightarrow R_{1}, \frac{1}{a} R_{2} \rightarrow R_{2} \\
-\frac{1}{a} R_{3} \rightarrow R_{3}, \frac{1}{2} R_{3}+R_{2} \rightarrow R_{2} \\
R_{1}-R_{2} \rightarrow R_{1} .
\end{array}
\]
Determine the elementary matrices corresponding corresponding to each the EROs given above, in the order in which they are given.
Solution
\[
\begin{array}{l}
E_{1}= \\
E_{2}= \\
E_{3}= \\
E_{4}= \\
E_{5}= \\
E_{6}=
\end{array}
\]
