TITLE: Matrices and power series
QUESTION [6 upvotes]: For any square matrix $A$, we can define $\sin A$ using the formal power series as follows:
\begin{equation}
\sin A=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}A^{2n+1}
\end{equation}
Prove or disprove: there exists a $2\times2$ real matrix $A$ such that
\begin{equation}
\sin A=\begin{bmatrix}1 & 2020 \\0& 1\end{bmatrix}
\end{equation}

REPLY [4 votes]: The relation $\, \sin(A)^2+\cos(A)^2=I$ also hold for matrices
So we would have to find a matrix for cosine such that $C^2=\begin{pmatrix}0&-4040\\0&0\end{pmatrix}$ which is not possible.