Answer
nonsingular.
Work Step by Step
Since one can show that
$$\left|\begin{array}{rrr} {\frac{1}{2}} & {\frac{3}{2}}&{2} \\ {\frac{2}{3}} & {-\frac{1}{3}}&{0}\\{1}&{1}&{1}\end{array}\right|=\frac{5}{6},$$
then the matrix
$$\left[\begin{array}{rrr} {\frac{1}{2}} & {\frac{3}{2}}&{2} \\ {\frac{2}{3}} & {-\frac{1}{3}}&{0}\\{1}&{1}&{1}\end{array}\right]$$
is nonsingular.
