Problem

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

$$\begin{aligned} 5x + ay =&\, 3,\\ 20x - 5y =&\, 2,\\ 2x - 3y + z =&\, 1, \end{aligned}$$

has a unique solution and interpret this situation geometrically.

(b) Given that $a = -\frac{5}{4}$, show that the system of equations in part (a) is inconsistent and interpret this situation geometrically.

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

$$\mathbf{A} = \begin{pmatrix} 5 & 0 & 0\\ 20 & -5 & 0\\ 2 & -3 & 1 \end{pmatrix}.$$

(c) Find a matrix $\mathbf{P}$ and a diagonal matrix $\mathbf{D}$ such that $(\mathbf{A} + 3\mathbf{I})^2 = \mathbf{P}\mathbf{D}\mathbf{P}^{-1}$.