<s>[INST] Differential equation involving matrices I want to show that  
$ Y(t)=\cos(At)$ and $Y(t)=\sin(At)$ satisfy the equation 
$$Y''+A^2Y=0 $$
subjected to the initial conditions $Y(0)=I, Y'(0)=0$ and $Y(0)=0, Y'(0)=I$ respictively
where $A$ is an constant $n \times n$ matrix..
 [/INST] I found the shifting of indices to be highly confusing so I propose a method that does not rely on that.
We will first show that $Y(t)=\cos (At)$ solves the initial value problem $Y''+A^2 Y=0$ where $Y(0)=I$ and $Y'(0)=0$. Because $\cos At$ is defined as the infinite sum $\sum\limits_{k=0}^{\infty} (-1)^{k}\frac{A^{2k}t^{2k}}{(2k)!}$ and differentiation is linear we can differentiate term by term to get:
\begin{eqnarray*}
Y''(t)&=&\frac{d^2}{dt^2}\left[    I -\frac{A^2t^2}{2!}+\frac{A^4t^4}{4!}-\frac{A^6t^6}{6!}+\ldots   \right]\\
&=&\left[  -2\frac{A^2}{2!} +4\cdot 3 \frac{A^4 t^2}{4!}  -6\cdot 5 \frac{A^6 t^4}{6!} +\ldots \right]\\
&=&-A^2\left[    I -\frac{A^2t^2}{2!}+\frac{A^4t^4}{4!}-\frac{A^6t^6}{6!}+\ldots   \right]\\
&=& -A^2 \sum\limits_{k=0}^{\infty} (-1)^k \frac{A^{2k}t^{2k} }{(2k)!}=-A^2Y(t)
\end{eqnarray*}
So $Y(t)$ satisfies $ Y''+A^2Y=0$. Also $Y(0)=\cos(0)=\sum\limits_{k=0}^{\infty} (-1)^{k}\frac{0^{2k}}{(2k)!}=I$ (The $0^{2k}$ in the summation is the zero matrix $[A\cdot t]_{t=0}=0$  and $0^0=I$, all higher powers of $0$ are of course $0$). And for $Y'(0)$ we find
\begin{eqnarray*}
Y'(0)&=&\frac{d}{dt}\left[  I -\frac{A^2t^2}{2!}+\frac{A^4t^4}{4!}-\frac{A^6t^6}{6!}+\ldots   \right] _{t=0}\\
&=& \left[   -2\frac{A^20}{2!}+4\frac{A^40^3}{4!}-6\frac{A^60^5}{6!}+\ldots   \right] =0
\end{eqnarray*}
(here the $0^k$ is just the scalar $t=0$ )
For $Y(t)=\sin(At)$ with $\sin(At):=\sum\limits_{k=0}^{\infty} (-1)^{k}\frac{A^{2k+1}t^{2k+1}}{(2k+1)!}$ we have: 
\begin{eqnarray*}
Y''(t)&=&\frac{d^2}{dt^2}\left[    At -\frac{A^3t^3}{3!}+\frac{A^5t^5}{5!}-\frac{A^7t^7}{7!}+\ldots   \right]\\
&=&\left[  -3\cdot 2\frac{A^3t}{3!} +5\cdot 4 \frac{A^5 t^3}{5!}  -5\cdot 6 \frac{A^7 t^5}{7!} +\ldots \right]\\
&=&-A^2\left[    At -\frac{A^3t^3}{3!}+\frac{A^5t^5}{5!}-\frac{A^7t^7}{7!}+\ldots   \right]\\
&=& -A^2 \sum\limits_{k=0}^{\infty} (-1)^{k}\frac{A^{2k+1}t^{2k+1}}{(2k+1)!}=-A^2Y(t)
\end{eqnarray*}
So $Y(t)=\sin(At)$ also satisfies $Y''(t)+A^2Y(t)=0$. Here however $Y(0)=\sin( 0)=\sum\limits_{k=0}^{\infty} (-1)^{k}\frac{0^{2k+1}}{(2k+1)!}=0$ (again using $0$ as the zero matrix in the summation, and now the lowest power of $0$ is $0^1$ so here we have an infinite sum of zeros). And
\begin{eqnarray}
Y'(0)&=&\frac{d}{dt}\left[    At -\frac{A^3t^3}{3!}+\frac{A^5t^5}{5!}-\frac{A^7t^7}{7!}+\ldots   \right] _{t=0}\\
&=& \left[  A -3\frac{A^30^2}{3!}+5\frac{A^50^4}{5!}-7\frac{A^70^6}{7!}+\ldots   \right] =A
\end{eqnarray}
I am guessing this is what you meant because even if we consider a scalar $A$ then $\left[\frac{d}{dt}\sin(At)\right]_{t=0}=A\cos(0)=A$.
</s>