Given that \(z = \cos \theta + \mathrm { j } \sin \theta\), express \(z ^ { n } + z ^ { - n }\) and \(z ^ { n } - z ^ { - n }\) in simplified trigonometrical form.
By considering \(\left( z + z ^ { - 1 } \right) ^ { 6 }\), show that
$$\cos ^ { 6 } \theta = \frac { 1 } { 32 } ( \cos 6 \theta + 6 \cos 4 \theta + 15 \cos 2 \theta + 10 )$$
Obtain an expression for \(\cos ^ { 6 } \theta - \sin ^ { 6 } \theta\) in terms of \(\cos 2 \theta\) and \(\cos 6 \theta\).
The complex number \(w\) is \(8 \mathrm { e } ^ { \mathrm { j } \pi / 3 }\). You are given that \(z _ { 1 }\) is a square root of \(w\) and that \(z _ { 2 }\) is a cube root of \(w\). The points representing \(z _ { 1 }\) and \(z _ { 2 }\) in the Argand diagram both lie in the third quadrant.
Find \(z _ { 1 }\) and \(z _ { 2 }\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\). Draw an Argand diagram showing \(w , z _ { 1 }\) and \(z _ { 2 }\).
Find the product \(z _ { 1 } z _ { 2 }\), and determine the quadrant of the Argand diagram in which it lies.
Show that the characteristic equation of the matrix
$$\mathbf { M } = \left( \begin{array} { r r r }
1 & - 4 & 5
2 & 3 & - 2
- 1 & 4 & 1
\end{array} \right)$$
is \(\lambda ^ { 3 } - 5 \lambda ^ { 2 } + 28 \lambda - 66 = 0\).
Show that \(\lambda = 3\) is an eigenvalue of \(\mathbf { M }\), and determine whether or not \(\mathbf { M }\) has any other real eigenvalues.
Find an eigenvector, \(\mathbf { v }\), of unit length corresponding to \(\lambda = 3\).
State the magnitude of the vector \(\mathbf { M } ^ { n } \mathbf { v }\), where \(n\) is an integer.
Using the Cayley-Hamilton theorem, obtain an equation for \(\mathbf { M } ^ { - 1 }\) in terms of \(\mathbf { M } ^ { 2 } , \mathbf { M }\) and \(\mathbf { I }\).