Problem

(a) Find the values of $k$ for which the system of equations

$$\begin{aligned} x - y + 2kz &= 1,\\ kx + y + 2z &= 2,\\ 2x - y + z &= 3, \end{aligned}$$

does not have a unique solution.

(b) Given that $k = -\frac{1}{2}$, show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically.

(c) Given instead that $k = -1$, show that the system of equations in part (a) is also inconsistent. Interpret this situation geometrically.

The matrix $\mathbf{A}$ is given by

$$\mathbf{A} = \begin{pmatrix} 1 & -1 & -2\\ -1 & 1 & 2\\ 2 & -1 & 1 \end{pmatrix}.$$

(d) Use the characteristic equation of $\mathbf{A}$ to show that $\mathbf{A}^4 = p\mathbf{A}^2 + q\mathbf{A}$ where $p$ and $q$ are integers to be determined.