Answer
See below
Work Step by Step
Given $$ A=\begin{bmatrix}
e^{2t} & e^{3t} & e^{-4t} \\ 2e^{2t} & 3e^{3t} & -4e^{-4t} \\ 4e^{2t} & 9e^{3t} & 16e^{-4t}
\end{bmatrix}$$
So, we get
$det (A)=\begin{vmatrix}
e^{2t} & e^{3t} & e^{-4t} \\ 2e^{2t} & 3e^{3t} & -4e^{-4t} \\ 4e^{2t} & 9e^{3t} & 16e^{-4t}
\end{vmatrix}\\
=e^{2t}.3e^{3t}.16e^{-4t}+e^{3t}.(-4e^{-4t}).(4e^{2t})+e^{-4t}2e^{2t}.9e^{3t}-e^{-4t}.3e^{3t}.4e^{2t}-e^{2t}.(-4e^{-4t}).9e^{3t}-e^{3t}.2e^{2t}.16e^{-4t}\\
=48e^t-16e^t+18e^t-12e^t+36e^t-32e^t\\
=42e^t$
