Question

Given
$A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$,
the state transition matrix $e^{At}$ is given by

• A. $\left[\begin{array}{cc}0& e^{-t}\\ e^{-t}& 0\end{array}\right]$
• B. $\left[\begin{array}{cc}{e}^t& 0\\ 0& e^t\end{array}\right]$
• C. $\left[\begin{array}{cc}{e}^{-t}& 0\\ 0& e^{-t}\end{array}\right]$
• D. $\left[\begin{array}{cc}0& e^t\\ e^t& 0\end{array}\right]$

Answer 1

The correct option is B $\left[\begin{array}{cc}{e}^t& 0\\ 0& e^t\end{array}\right]$

$[sI-A]=\left[\begin{array}{cc}s& 0\\ 0& s\end{array}\right]-\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$[sI-A]=\left[\begin{array}{cc}s-1& 0\\ 0& s-1\end{array}\right]$
$e^{At}=[sI-A{]}^{-1}$
$=\left[\begin{array}{cc}\frac{1}{s-1}& 0\\ 0& \frac{1}{s-1}\end{array}\right]=\left[\begin{array}{cc}{e}^t& 0\\ 0& e^t\end{array}\right]$