8 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { c c c } t - 1 & t - 1 & t - 1
1 - t & 6 & t
2 - 2 t & 2 - 2 t & 1 \end{array} \right)\).
- Find, in fully factorised form, an expression for \(\operatorname { det } \mathbf { A }\) in terms of \(t\).
- State the values of \(t\) for which \(\mathbf { A }\) is singular.
You are given the following system of equations in \(x , y\) and \(z\), where \(b\) is a real number.
$$\begin{aligned}
\left( b ^ { 2 } + 1 \right) x + \left( b ^ { 2 } + 1 \right) y + \left( b ^ { 2 } + 1 \right) z & = 5
\left( - b ^ { 2 } - 1 \right) x + \quad 6 y + \left( b ^ { 2 } + 2 \right) z & = 10
\left( - 2 b ^ { 2 } - 2 \right) x + \left( - 2 b ^ { 2 } - 2 \right) y + \quad z & = 15
\end{aligned}$$ - Determine which one of the following statements about the solution of the equations is true.
- There is a unique solution for all values of \(b\).
- There is a unique solution for some, but not all, values of \(b\).
- There is no unique solution for any value of \(b\).