The set of all values of t ∈ ℝ , for which the matrix [ cases e ^ t...

Let

A = \left[\begin{cases}{e}^{t} & {e}^{- t}\left(\sin t - 2\cos t\right) & {e}^{- t}\left(- 2\sin t - \cos t\right) \\ {e}^{t} & {e}^{- t}\left(2\sin t + \cos t\right) & {e}^{- t}\left(\sin t - 2\cos t\right) \\ {e}^{t} & {e}^{- t}\cos t & {e}^{- t}\sin t\end{cases}\right]

Given matrix $A$ is invertible if

\left|A\right| \ne 0$</p><p>$⇒ \left|$$\begin{array}{ll}{e}^{t} {e}^{- t}\left(\sin t - 2\cos t\right) {e}^{- t}\left(- 2\sin t - \cos t\right) \\ {e}^{t} & {e}^{- t}\left(2\sin t + \cos t\right) & {e}^{- t}\left(\sin t - 2\cos t\right) \\ {e}^{t} & {e}^{- t}\cos t & {e}^{- t}\sin t\end{array}$$\right| \ne 0$</p><p>$⇒ {e}^{t} \cdot {e}^{- t} \cdot {e}^{- t}\left|$\begin{cases}1 & \sin t - 2\cos t & - 2\sin t - \cos t \\ 1 & 2\sin t + \cos t & \sin t - 2\cos t \\ 1 & \cos t & \sin t\end{cases}$\right| \ne 0$</p><p>Applying${R}_{1} \to {R}_{1} - {R}_{2} then ${R}_{2} \to {R}_{2} - {R}_{3}$, ee get

{e}^{- t}\left|\begin{cases}0 & - \sin t - 3\cos t & - 3\sin t + \cos t \\ 0 & 2\sin t & - 2\cos t \\ 1 & \cos t & \sin t\end{cases}\right| \ne 0

By expanding we have,

${e}^{- t} \times \left(2\sin t \cos t + 6\cos^{2}t + 6\sin^{2}t - 2\sin t \cos t\right) \ne 0$

$⇒ {e}^{- t} \times 6 \ne 0$ for $\forall t \in \mathbb{R}$