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3 and 4 .Determinants and Matrices
hard
If $A\, = \,\left[ {\begin{array}{*{20}{c}}
{{e^t}}&{{e^{ - t}}\,\cos \,t}&{{e^{ - t}}\,\sin \,t}\\
{{e^t}}&{ - {e^{ - t}}\,\cos \, - {e^{ - t}}\,\sin \,t}&{ - {e^{ - t}}\,\sin \,t\, + \,{e^{ - t}}\,\cos \,t}\\
{{e^t}}&{2{e^{ - t}}\,\sin \,t}&{2{e^{ - t}}\,\cos \,t}
\end{array}} \right]$ Then $A$ is
A
Invertible only if $t = \frac {\pi }{2}$
B
not invertible for any $t \in R$
C
invertible for all $t \in R$
D
invertible only if $t = \pi $
(JEE MAIN-2019)
Solution
$\left| A \right| = {e^{ – t}}\left| {\begin{array}{*{20}{c}}
1&{\cos \,t}&{\sin \,t}\\
1&{ – \cos \,t – \sin \,t}&{\, – \sin \,t + \cos \,t}\\
1&{2\sin \,t}&{ – 2\cos \,t}
\end{array}} \right|$
$ = {e^{ – t}}\left[ {5{{\cos }^2}t + 5{{\sin }^2}t} \right]\forall t \in R$
$ = 5{e^{ – t}} \ne 0\forall t \in R$
Standard 12
Mathematics
