Question

If $A = \begin{bmatrix}e^{t}&e^{t} \cos t&e^{-t}\sin t\\\ e^{t}&-e^{t} \cos t -e^{-t}\sin t&-e^{-t} \sin t+ e^{-t} \cos t\\\ e^{t}&2e^{-t} \sin t&-2e^{-t} \cos t\end{bmatrix}$ Then $A$ is -

• A. Invertible only if $t = \frac{\pi}{2}$
• B. not invertible for any $t \epsilon R$
• C. invertible for all $t \epsilon R$
• D. invertible only if $t = \pi$

Answer 1

Answer: (C)

invertible for all $t \epsilon R$

Answer 2

$\left|A\right| = e^{-t} \begin{vmatrix}1&\cos t&\sin t\\\ 1&-\cos t -\sin t&-\sin t + \cos t\\\ 1&2\sin t&-2 \cos t\end{vmatrix}$
$= e^{-t} \left[5 \cos^{2} t + 5\sin^{2 }t\right] \forall t \in R$
$= 5e^{-t} \ne0 \forall t \in R$