2. Consider the system of linear equations
$$\begin{array}{r}2y+3z=0\\ x+2y+z=0\\ -2x-2y+z=0\end{array}$$
a) [6 marks Solve the system using Gaussian or Gauss-Jordan elimination.
b) $[6$ marks $]$ Write the system in matrix form $AX=B$. Find det $A$ by cofactor expansion.
c) [2 marks] Is the system consistent? Is the system homogeneous?

Step 1 ：Given the system of linear equations: $$\begin{array}{r}2y+3z=0\\ x+2y+z=0\\ -2x-2y+z=0\end{array}$$

Step 2 ：Write this system in augmented matrix form: $$\begin{array}{cccc}0& 2& 3& 0\\ 1& 2& 1& 0\\ -2& -2& 1& 0\end{array}$$

Step 3 ：Perform row operations to bring it to reduced row-echelon form: $$\begin{array}{cccc}1& 0& -2& 0\\ 0& 1& 3/2& 0\\ 0& 0& 0& 0\end{array}$$

Step 4 ：This corresponds to the system of equations: $$\begin{array}{r}x-2z=0\\ y+3/2z=0\\ 0=0\end{array}$$

Step 5 ：From the first equation, express x in terms of z: $x=2z$

Step 6 ：From the second equation, express y in terms of z: $y=-3/2z$

Step 7 ：Since there is no equation for z, let z be a free variable

Step 8 ：The solution to the system of equations is $x=2z$, $y=-3/2z$, z is free

Step 9 ：Final Answer: The solution to the system of equations is $\boxed{{\displaystyle x=2z,y=-3/2z,z\text{ is free}}}$