Prove the following

Question:

If 

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

Then $\mathrm{A}$ is-

  1. Invertible only if $t=\frac{\pi}{2}$

  2. not invertible for any $t \varepsilon R$

  3. invertible for all $t \varepsilon R$

  4. invertible only if $t=\pi$

Correct Option: , 3

Solution:

$|A|=e^{-t}\left|\begin{array}{ccc}1 & \cos t & \sin t \\ 1 & -\cos t-\sin t & -\sin t+\cos t \\ 1 & 2 \sin t & -2 \cos t\end{array}\right|$

$=\mathrm{e}^{-\mathrm{t}}\left[5 \cos ^{2} \mathrm{t}+5 \sin ^{2} \mathrm{t}\right] \forall \mathrm{t} \in \mathrm{R}$

$=5 \mathrm{e}^{-\mathrm{t}} \neq 0 \forall \mathrm{t} \in \mathrm{R}$

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