Taylor series of functions with matrix input

Today my professor said something interesting, replace $x$ in $f(x)$ with matrix \begin{bmatrix}x&1\\0&x\end{bmatrix}

Then $$f(\begin{bmatrix}x&1\\0&x\end{bmatrix}) = f(x)\begin{bmatrix}1&0\\0&1\end{bmatrix} + f^{'}(x)\begin{bmatrix}0&-1\\1&0\end{bmatrix}$$

and asked us to find the higher order terms ($f^{''}(x), f^{'''}(x) ....$) and extend it to multi variable functions.

I didn't understand how he came up with this.

After some googling, this seems similar to matrix exponential from lie groups.

from Hot Weekly Questions - Mathematics Stack Exchange