The matrix \(\mathbf { X }\) is given by \(\mathbf { X } = \left( \begin{array} { r r r } a & 3 & - 2 0 & a & 5 1 & 2 & 1 \end{array} \right)\). Show that the determinant of \(\mathbf { X }\) is \(a ^ { 2 } - 8 a + 15\).
Explain briefly why the equations
$$\begin{array} { r }
3 x + 3 y - 2 z = 1
3 y + 5 z = 5
x + 2 y + z = 2
\end{array}$$
do not have a unique solution and determine whether these equations are consistent or inconsistent.
Use an algebraic method to find the square roots of the complex number \(9 + 40 \mathrm { i }\).
Show that \(9 + 40 \mathrm { i }\) is a root of the quadratic equation \(z ^ { 2 } - 18 z + 1681 = 0\).
By using the substitution \(z = \frac { 1 } { u ^ { 2 } }\), find the roots of the equation \(1681 u ^ { 4 } - 18 u ^ { 2 } + 1 = 0\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.