10 You are given the matrix \(\mathbf { A }\) where \(\mathbf { A } = \left( \begin{array} { l l l } a & 2 & 0
0 & a & 2
4 & 5 & 1 \end{array} \right)\).
- Find, in terms of \(a\), the determinant of \(\mathbf { A }\), simplifying your answer.
- Hence find the values of \(a\) for which \(\mathbf { A }\) is singular.
You are given the following equations which are to be solved simultaneously.
$$\begin{aligned}
a x + 2 y & = 6
a y + 2 z & = 8
4 x + 5 y + z & = 16
\end{aligned}$$ - For each of the values of \(a\) found in part (b) determine whether the equations have
- a unique solution, which should be found, or
- an infinite set of solutions or
- no solution.