1. ODE Matrix exponential

Identify the following using known results about the matrix exponential.

e. Suppose the pair (x(t), y(t)) is a solution to an autonomous first order system whose matrix exponential is e^(tA).
If (x(3),y(3)) = (5,7), how can (x(5),y(5)) be expressed?????????

2. Recall that the first-order linear system of ODE given by $X'=AX$, where $X(t)=\begin{bmatrix}x(t)\\y(t)\end{bmatrix}$ is the (column) vector of our multivariable solution, has solution given by the matrix exponential $X(t)=e^{tA}X_0$, where $X_0=\begin{bmatrix}x(0)\\y(0)\end{bmatrix}$ is our initial value. So we have in our problem that $X(3)=e^{3A}X_0=\begin{bmatrix}5\\7\end{bmatrix}$.
Matrix exponentials have the property $e^{sA}e^{tA}=e^{(s+t)A}$, so
$X(5)=e^{5A}X_0$
$\;=e^{2A}e^{3A}X_0$
$\;=e^{2A}\begin{bmatrix}5\\7\end{bmatrix}$.

--Kevin C.