$(1)$Trace of the matrix $A=$Sum of the leading diagonal element (main diagonal sometimes principal diagonal, primary diagonal)$=$Sum of all Eigen-values.
$(2)$Product of all Eigen values$=Det(A)=|A|$
Suppose $\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4},\lambda_{5}$
$(1)$Trace of the matrix$= \lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4}+\lambda_{5}$
$(2) \lambda_{1}.\lambda_{2}.\lambda_{3}.\lambda_{4}.\lambda_{5}=|A|$
In given question$:$
In case singular matrix $|A|_{5\times5}=0$
$\bullet$The complex conjugate of $a+ib$ is $a-ib$ where $i=\sqrt{-1}$
$\bullet$The conjugate of $a+b\sqrt{n}$ is $a-b\sqrt{n}$
Given that
$\lambda_{1}=2+3\sqrt{-2}=2+3\sqrt{-1}.\sqrt{2}=2+3\sqrt{2}i$
$\lambda_{2}=2-3\sqrt{2}i$
$\lambda_{3}=2-3\sqrt{2}$
$\lambda_{4}=2+3\sqrt{2}$
$\lambda_{5}=0$
For many non trivial solutions $AX=0 (|A|=0)$
So$,(1)$ and $(3)$ are right.