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Mathematics Question on Matrices

The set of all values of t∈Rt∈R, for which the matrix [ete−t(sin⁡ t−2cos ⁡t)e−t(−2sin⁡ t−cos ⁡t)\[0.3em]ete−t(2sin ⁡t+cos ⁡t)e−t(sin ⁡t−2cos ⁡t)\[0.3em]ete−tcos ⁡te−tsin ⁡t]\begin{bmatrix} e^t & e^{−t}(sin⁡\ t−2cos\ ⁡t) & e^{−t}(−2sin⁡\ t−cos\ ⁡t) \\\[0.3em] e^t & e^{−t}(2sin\ ⁡t+cos\ ⁡t) & e^{−t}(sin\ ⁡t−2cos\ ⁡t) \\\[0.3em] e^t & e^{−tcos\ ⁡t}& e^{−tsin\ ⁡t} \end{bmatrix} is invertible, is

A

(2k+1)π2,k∈Z \\{(2k+1)\frac \pi2 ,k∈Z\\}

B

R

C

kπ+π4,k∈Z\\{k\pi+\frac \pi4 ,k∈Z\\}

D

kπ,k∈Z\\{k\pi,k∈Z\\}

Answer

(2k+1)π2,k∈Z \\{(2k+1)\frac \pi2 ,k∈Z\\}

Explanation

Solution

The Correct Option is (B): R