Answer
false
Work Step by Step
We know that the determinant of a matrix \[
\left[\begin{array}{rr}
a & b \\
c &d \\
\end{array} \right]
\]
is $D=ad-bc$.
Also, a matrix is non-invertible if $D=0$.
here $D=x^2y-xy^2=xy(y-x)$.
Thus if it is non-invertible then $x=0$, $y=0$ or $x=y$, thus the statement is false.
