Problem

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

$$\begin{aligned} 6x+ay&=3,\\ 2x-y&=1,\\ x+5y+4z&=2 \end{aligned}$$

has a unique solution.

(b) Show that the system of equations in part (a) is consistent for all values of $a$.

The matrix $A$ is given by

$$A=\begin{pmatrix} 6&0&0\\ 2&-1&0\\ 1&5&4 \end{pmatrix}.$$

(c) Find a matrix $P$ and a diagonal matrix $D$ such that

$$(14A+24I)^2=PDP^{-1}.$$

(d) Use the characteristic equation of $A$ to show that

$$(14A+24I)^2=A^4(A+bI)^2,$$

where $b$ is an integer to be determined.