CAIE A-Level Further Math A2 2.2.6 Eigen Terminology Questions

$(\lambda-a)(\lambda+1)(\lambda+4)=\lambda^{3}+(5-a) \lambda^{2}+(4-5 a) \lambda-4 a=0$
Finds characteristic equation.

$4 a \mathbf{A}^{-1}=\mathbf{A}^{2}+(5-a) \mathbf{A}+(4-5 a) \mathbf{I}$
Multiplies through by $\mathbf{A}^{-1}$.

$\mathbf{A}^{2}=\left(\begin{array}{ccc}a^{2} 2 a^{2}-17 a^{2}-1 0 16 0 -15 1\end{array}\right)$$4 a \mathbf{A}^{-1}=\left(\begin{array}{ccc}a^{2} 2 a^{2}-17 a^{2}-1 0 16 0 -15 1\end{array}\right)+(5-a)\left(\begin{array}{ccc}a 2 a+5 a+1 0 -4 0 3 -1\end{array}\right)+(4-5 a)\left(\begin{array}{lll}1 0 0 1 0 0 1\end{array}\right)$
Substitutes for and A in correct equation.

$\mathbf{A}^{-1}=\frac{1}{4 a}\left(\begin{array}{ccc}4 5 a+8 4 a+4 0 -a 0 -3 a -4 a\end{array}\right)$
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