I've been given an identity (that I don't know how to prove unfortunately), and been asked to use it to compute exp$(xM)$, where $$ M = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 &1 & 1 \\ 1 &1 & 1 & 1 \\ 1 &1 & 1 & 1 \end{bmatrix} $$ The identity is exp$(M) = I_n + P(QP)^{-1}($exp$(QP) - I_k)Q$.
$M$ is $n \times n$, $P$ is $n \times k$, $Q$ is $k \times n$. $M = PQ$.
Can anyone give me any pointers?
I worked out that $$P = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix}, Q = \begin{bmatrix} 1 & 1 & 1 & 1 \end{bmatrix} $$ using that the rank of $M$ is $1$, and filling in the identity to get
$$ exp(xM) = \dfrac{1}{4} \begin{bmatrix} 3 + e^{4x} & -1 + e^{4x} & -1 + e^{4x} & -1 + e^{4x} \\ -1 + e^{4x} & 3 + e^{4x} & -1 + e^{4x}& -1 + e^{4x} \\ -1 + e^{4x} & -1 + e^{4x} & 3 + e^{4x} & -1 + e^{4x} \\ -1 + e^{4x}& -1 + e^{4x} & -1 + e^{4x} & 3 + e^{4x} \end{bmatrix} $$