Let $A,B,C \in \operatorname{Mat}_2(\mathbb{R})$ define the real matrices: $$A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \\ \end{pmatrix},\ B =\begin{pmatrix} 0 & 0 \\ 0 & 1 \\ \end{pmatrix},\ C = A+B=\begin{pmatrix} 0 & 1 \\ 0 & 1 \\ \end{pmatrix} $$ Show that $$\exp(A;t) = \begin{pmatrix} 1 & t \\ 0 & 1 \\ \end{pmatrix}\text{ and } \exp(B;t)=\begin{pmatrix} 1 & 0 \\ 0 & e^t \\ \end{pmatrix} \text{ for } t\in \mathbb{R}$$ and show that $$\exp(C;t)=\begin{pmatrix} 1 & e^t-1 \\ 0 & e^t \\ \end{pmatrix}\text{ for } t\in \mathbb{R}$$
I am a bit unsure with the notation used here for $\exp(A;t)=\begin{pmatrix} 1 & t \\ 0 & 1 \\ \end{pmatrix}$ and how am I supposed to show this based on the information for the matrices $A,B,C$ ?
I assume that $e^A = I + A + {1 \over 2!} A^2+ \cdots$.
Note that $A^2 = 0$, hence $e^{At} = I + tA$.
Note that $B^k = B$ for all $k \ge 1$, hence $e^{Bt} = I + B(t + {t^2 \over 2} + \cdots) )= \begin{bmatrix} 1 & 0 \\ 0 & e^t \end{bmatrix}$.
Finally, as Jean-Claude noted, $C^k = C$ for $k \ge 1$.