Question

If $\left[\begin{array}{ccc}{e}^t& e^{-t}\cos t& e^{-t}\sin t\\ e^t& -e^{-t}\cos t-e^{-t}\sin t& -e^{-t}\sin t+e^{-t}\cos t\\ e^t& 2e^{-t}\sin t& -2e^{-t}\cos t\end{array}\right]$ 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 $t\epsilon R$

$|A|=e^{-t}\left[\begin{array}{ccc}1& \cos t& \sin t\\ 1& -\cos t-\sin t& -sint+\cos t\\ 1& 2\sin t& -2\cos t\end{array}\right]$
$=e^{-t}[5{\cos }^{2}t+5{\sin }^{2}t]\forall t\in R$
$=5e^{-t}\ne 0\forall t\in R$