GATE 2018 | MATHS | Q-55

Question

Let \( A \) be a \(3 \times 3\) matrix with real entries. If three solutions of the linear system of differential equations \(\dot{x}(t) = Ax(t)\) are given by
\[
\begin{bmatrix}
e^t - e^{2t} \\
-e^{t} + e^{2t} \\
e^t + e^{2t}
\end{bmatrix},
\begin{bmatrix}
-e^{2t} - e^{-t} \\
e^{2t} - e^{-t} \\
e^{2t} + e^{-t}
\end{bmatrix},
\]
and
\[
\begin{bmatrix}
e^{-t} + 2e^t \\
e^{-t} - 2e^t \\
-e^{-t} + 2e^t
\end{bmatrix},
\]
then the sum of the diagonal entries of \( A \) is equal to