6×6 Matrix Exponential using the Cayley-Hamilton theorem

Question

For the $6×6$ matrix $$P= \begin{pmatrix} 0 & 0 & 0 & 0 & 2 & 0 \\ -1& 0 & 0 & 4 & 2 & 0 \\ 0 & 1 &-1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 &-1 & 0 & 0 & 1 \\ \end{pmatrix} $$ What is the answer to $e^{Pt}$ using the Cayley-Hamilton theorem?
The characteristic polynomial is $x^6$.

Answer 1

From Cayley-Hamilton we get $P^6=0,$ hence $P^n=0$ for $n \ge 6.$ Then

$$e^{Pt}= \sum_{k=0}^5\frac{t^kP^k}{k!}.$$