10 The matrix \(\mathbf { A }\) is defined by
$$\mathbf { A } = \left( \begin{array} { r r r }
1 & 5 & 1
1 & - 2 & - 2
2 & 3 & \theta
\end{array} \right)$$
- (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
$$\begin{aligned}
x + 5 y + z & = - 1
x - 2 y - 2 z & = 0
2 x + 3 y + \theta z & = \theta
\end{aligned}$$ - Solve the system of equations when \(\theta \neq - 1\).
- Find the general solution when \(\theta = - 1\).
- 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.