63 Checking the solution

Here

sI-A=\begin{bmatrix}s&-1&0\\ 0&s&-1\\ -2&8&s+6\end{bmatrix};

computing only the relevant entries, we have

{\hbox{adj}}(sI-A)^{t}=\begin{bmatrix}*&*&*\\ *&*&*\\ 1&s&s^{2}\end{bmatrix}

C{\hbox{adj}}(sI-A)B=\begin{bmatrix}1&3&2\end{bmatrix}\begin{bmatrix}*&*&1\\ *&*&s\\ *&*&s^{2}\end{bmatrix}\begin{bmatrix}0\\ 0\\ 1\end{bmatrix}

=1+3s+2s^{2}