Cayley-Hamilton Block Matrix In Q4 d) I understand how to get the expression for A (using Q3), but I don't understand how you can then say that C is of the form described. 



Well, because $C$ is a block diagonal matrix, we have
$$ C^² + C + I_4 = \begin{pmatrix}
        A^² & 0  \\
        0 & A^²  \\
        \end{pmatrix} +  \begin{pmatrix}
        A & 0  \\
        0 & A  \\
        \end{pmatrix} + \begin{pmatrix}
        I_2 & 0  \\
        0 & I_2  \\
        \end{pmatrix}  $$
which simplifies to
$$
C^² + C + I_4 = \begin{pmatrix}
        A^² + A + I_2 & 0  \\
        0 & A^² + A + I_2  \\
        \end{pmatrix} = \begin{pmatrix}
        0 & 0  \\
        0 & 0  \\
        \end{pmatrix}
$$
