Let $A = \left( \begin{array} { c c }
2 & 0 \
0 & - 3
\end{array} \right)$ . Then $e ^ { A t } =$
A) $I + A t$
B) $I + A t + A ^ { 2 } t ^ { 2 } / 2$
C) $A t + A ^ { 2 } t ^ { 2 } / 2$
D) $\left( \begin{array} { c c }
e ^ { 2 t } & 0 \
0 & e ^ { - 3 t }
\end{array} \right)$
E) $\left( \begin{array} { c c }
e ^ { 2 t - 1 } - 1 & 0 \
0 & e ^ { - 3 t } - 1
\end{array} \right)$

Question

Let $A = \left( \begin{array} { c c } 
2 & 0 \
0 & - 3
\end{array} \right)$ . Then $e ^ { A t } =$
A) $I + A t$
B) $I + A t + A ^ { 2 } t ^ { 2 } / 2$
C) $A t + A ^ { 2 } t ^ { 2 } / 2$
D) $\left( \begin{array} { c c } 
e ^ { 2 t } & 0 \
0 & e ^ { - 3 t }
\end{array} \right)$
E) $\left( \begin{array} { c c } 
e ^ { 2 t - 1 } - 1 & 0 \
0 & e ^ { - 3 t } - 1
\end{array} \right)$

Quizplus · Accepted Answer

The matrix exponential $e^{At}$ for a diagonal matrix $A$ can be computed by taking the exponential of each diagonal element separately, resulting in a diagonal matrix where each diagonal entry is the exponential of the corresponding entry in $A$ multiplied by $t$.

Let A=(200−3)A = \left( \begin{array} { c c } 2 & 0 \\0 & - 3\end{array} \right)A=(20​0−3​)

Let $A = \left( \begin{array} { c c } 2 & 0 \\0 & - 3\end{array} \right)$