$$$e^{\left[\begin{array}{cc}t & - t\\0 & t\end{array}\right]}$$$

Find $$$e^{\left[\begin{array}{cc}t & - t\\0 & t\end{array}\right]}$$$.

First, diagonalize the matrix (for steps, see matrix diagonalization calculator).

Since the matrix is not diagonalizable, write it as the sum of the diagonal matrix $$$D = \left[\begin{array}{cc}t & 0\\0 & t\end{array}\right]$$$ and the nilpotent matrix $$$N = \left[\begin{array}{cc}0 & - t\\0 & 0\end{array}\right]$$$.

Notice that $$$N^{2} = \left[\begin{array}{cc}0 & 0\\0 & 0\end{array}\right]$$$.

This means that $$$e^{N} = I + N$$$, i.e. $$$e^{\left[\begin{array}{cc}0 & - t\\0 & 0\end{array}\right]} = \left[\begin{array}{cc}1 & 0\\0 & 1\end{array}\right] + \left[\begin{array}{cc}0 & - t\\0 & 0\end{array}\right] = \left[\begin{array}{cc}1 & - t\\0 & 1\end{array}\right].$$$

The exponential of a diagonal matrix is a matrix whose diagonal entries are exponentiated: $$$e^{\left[\begin{array}{cc}t & 0\\0 & t\end{array}\right]} = \left[\begin{array}{cc}e^{t} & 0\\0 & e^{t}\end{array}\right].$$$

Now, $$$e^{\left[\begin{array}{cc}t & - t\\0 & t\end{array}\right]} = e^{\left[\begin{array}{cc}t & 0\\0 & t\end{array}\right] + \left[\begin{array}{cc}0 & - t\\0 & 0\end{array}\right]} = e^{\left[\begin{array}{cc}t & 0\\0 & t\end{array}\right]}\cdot e^{\left[\begin{array}{cc}0 & - t\\0 & 0\end{array}\right]} = \left[\begin{array}{cc}e^{t} & 0\\0 & e^{t}\end{array}\right]\cdot \left[\begin{array}{cc}1 & - t\\0 & 1\end{array}\right].$$$

Finally, multiply the matrices:

$$$\left[\begin{array}{cc}e^{t} & 0\\0 & e^{t}\end{array}\right]\cdot \left[\begin{array}{cc}1 & - t\\0 & 1\end{array}\right] = \left[\begin{array}{cc}e^{t} & - t e^{t}\\0 & e^{t}\end{array}\right]$$$ (for steps, see matrix multiplication calculator).

$$$e^{\left[\begin{array}{cc}t & - t\\0 & t\end{array}\right]} = \left[\begin{array}{cc}e^{t} & - t e^{t}\\0 & e^{t}\end{array}\right]$$$A