Answer
See below
Work Step by Step
Given $$ A=\begin{bmatrix}
e^{-t} & e^{-5t} & e^{2t} \\ -e^{-t} & -5e^{-5t} & 2e^{2t} \\ e^{-t} & 25e^{-5t} & 4e^{2t}
\end{bmatrix}$$
So, we get
$det (A)=\begin{vmatrix}
e^{-t} & e^{-5t} & e^{2t} \\ -e^{-t} & -5e^{-5t} & 2e^{2t} \\ e^{-t} & 25e^{-5t} & 4e^{2t}
\end{vmatrix}\\
=e^{-t}(-5e^{-5t}).4e^{2t}+e^{-5t}.2e^{2t}.e^{-t}+e^{2t}.(-e^{-t}).25e^{-5t}-e^{-t}.(-5e^{-5t}).e^{2t}-25e^{-5t}.2e^{2t}.e^{-t}-4e^{2t}.(-e^{-t}).e^{-5t}\\
=-20e^{-4t}+2e^{-4t}-25e^{-4t}+e^{-4t}-50e^{-4t}+4e^{-4t}\\
=-84e^{-4t}$
