The set of all values of tR for which the matrix [pmatrixe^t & e^-t( t-2 t) & e^-t(-2 t- t…

Question

The set of all values of $t\in\mathbb{R}$ for which the matrix
$$\begin{pmatrix}e^t & e^{-t}(\sin t-2\cos t) & e^{-t}(-2\sin t-\cos t)\ e^t & e^{-t}(2\sin t+\cos t) & e^{-t}(\sin t-2\cos t)\ e^t & e^{-t}\cos t & e^{-t}\sin t\end{pmatrix}$$
is invertible, is

Accepted Answer

If the matrix is invertible then its determinant
should not be zero.

So, $$
\left|\[\begin{array}{ccc} e^t & e^{-t}(\sin t-2 \cos t) & e^{-t}(-2 \sin t-\cos t) \ e^t & e^{-t}(2 \sin t+\cos t) & e^{-t}(\sin t-2 \cos t) \ e^t & e^{-t} \cos t & e^{-t} \sin t \end{array}$$\right| \neq 0
\]

$$
\Rightarrow e^t \times e^{-t} \times e^{-t}\left|\[\begin{array}{ccc} 1 & \sin t-2 \cos t & -2 \sin t-\cos t \ 1 & 2 \sin t+\cos t & \sin t-2 \cos t \ 1 & \cos t & \sin t \end{array}$$\right| \neq 0
\]

Applying, $R_1 \rightarrow R_1-R_2$ then $R_2 \rightarrow R_2-R_3$,

$$
\begin{aligned}
& e^{-t}\left|\[\begin{array}{ccc} 0 & -\sin t-3 \cos t & -3 \sin t+\cos t \ 0 & 2 \sin t & -2 \cos t \ 1 & \cos t & \sin t \end{array}\end{aligned}$$\right| \neq 0 \\
\& \Rightarrow e\^{-t}\left(2 \sin t \cos t+6 \cos \^2 t+6 \sin \^2 t-2 \sin t \cos t\right) \neq 0 \\
\& \Rightarrow 6e\^{-t} \neq 0, \text { for } \forall t \in R .

\]

Solutions · 1