My task is to prove that:
If $x^{(1)},x^{(2)},x^{(3)}$ are solutions of $X'=A(t)X$ on some interval $I$, show that the Wronskian of $x^{(1)},x^{(2)},x^{(3)}$ is identically zero or else never vanishes.
After a long calculation I computed that the Wronskian is:
$e^{\int {(a_{11}+a_{22}+a_{33})dt}}$. Where $a_{11},a_{22},a_{33}$ are entries from the matrix $A$. My question is:
How from $e^{\int {(a_{11}+a_{22}+a_{33})dt}}$ the conclusion of my task follows?
I don't get it, any help is highly appreciated.
You have forgotten a mulitplicative term, that is, the term $\det\left(x^{\left(1\right)}\left(t_{0}\right),x^{\left(2\right)}\left(t_{0}\right),x^{\left(3\right)}\left(t_{0}\right)\right) $ where $t_{0}$ is the initial time given in your initial conditions. I let you check your calculus, and find your answer thanks to this last term.