Important properties of Eigen values:-
$(1)$Sum of all eigen values$=$Sum of leading diagonal(principle diagonal) elements=Trace of the matrix.
$(2)$ Product of all Eigen values$=Det(A)=|A|$
$(3)$ Any square diagonal(lower triangular or upper triangular) matrix eigen values are leading diagonal (principle diagonal)elements itself.
Example$:$$A=\begin{bmatrix} 1& 0& 0\\ 0&1 &0 \\ 0& 0& 1\end{bmatrix}$
Diagonal matrix
Eigenvalues are $1,1,1$
$B=\begin{bmatrix} 1& 9& 6\\ 0&1 &12 \\ 0& 0& 1\end{bmatrix}$
Upper triangular matrix
Eigenvalues are $1,1,1$
$C=\begin{bmatrix} 1& 0& 0\\ 8&1 &0 \\ 2& 3& 1\end{bmatrix}$
Lower triangular matrix
Eigenvalues are $1,1,1$
Now come to the question
$A=\begin{bmatrix} 2\\-4 \\7 \end{bmatrix}_{3\times1}.\begin{bmatrix} 1 &9 &5 \end{bmatrix}_{1\times3}$
$A=\begin{bmatrix} 2&18 &10 \\ -4&-36 &-20 \\7 &63 &35 \end{bmatrix}$
Now find $Det(A)=|A|=\begin{vmatrix} 2&18 &10 \\ -4&-36 &-20 \\7 &63 &35 \end{vmatrix}$
$|A|=2\times4\times7\begin{vmatrix} 1&9&5 \\ -1&-9 &-5 \\1 &9 &5 \end{vmatrix}$
$|A|=0$
so $xyz=0$