10 The matrix $\mathbf { A }$ is defined by

$$\mathbf { A } = \left( \begin{array} { r r r } 
1 & 5 & 1 \\
1 & - 2 & - 2 \\
2 & 3 & \theta
\end{array} \right)$$

(i) (a) Find the rank of $\mathbf { A }$ when $\theta \neq - 1$.\\

(b) Find the rank of $\mathbf { A }$ when $\theta = - 1$.\\

Consider the system of equations

$$\begin{aligned}
x + 5 y + z & = - 1 \\
x - 2 y - 2 z & = 0 \\
2 x + 3 y + \theta z & = \theta
\end{aligned}$$

(ii) Solve the system of equations when $\theta \neq - 1$.\\

(iii) Find the general solution when $\theta = - 1$.\\

(iv) Show that if $\theta = - 1$ and $\phi \neq - 1$ then $\mathbf { A } \mathbf { x } = \left( \begin{array} { r } - 1 \\ 0 \\ \phi \end{array} \right)$ has no solution.\\

\hfill \mbox{\textit{CAIE FP1 2019 Q10}}