Solve a differential equation involving matrices.
Solution 1:
\begin{align} e^{(t-s)A}=e^{-(t-s)}e^{t-s}e^{(t-s)A}=e^{-(t-s)}e^{(t-s)(A+I)}=e^{-(t-s)}e^{\left[\begin{array}{cc}0&t-s\\0&0\\\end{array}\right]}=e^{-(t-s)}\left(I+\left[\begin{array}{cc}0&t-s\\0&0\\\end{array}\right]\right)=e^{-(t-s)}\left[\begin{array}{cc}1&t-s\\0&1\\\end{array}\right] \end{align} where I have used formula $e^A=I+A$, which is valid if $A^2=0$.