Answer
$-\dfrac{11}{40}$
Work Step by Step
The determinant of a $2\times2$ matrix $A=\begin{bmatrix}a & b\\c&d
\end{bmatrix}$ is given by $ad-bc.$ Hence, the determinant of the given matrix, $
\begin{bmatrix}
\dfrac{1}{2}&\dfrac{2}{3}\\\\\dfrac{3}{5}&\dfrac{1}{4}
\end{bmatrix}$ is
\begin{align*}\require{cancel}
&
\dfrac{1}{2}\left(\dfrac{1}{4}\right)-\dfrac{3}{5}\left(\dfrac{2}{3}\right)
\\\\&=
\dfrac{1}{2}\left(\dfrac{1}{4}\right)-\dfrac{\cancel3}{5}\left(\dfrac{2}{\cancel3}\right)
\\\\&=
\dfrac{1}{8}-\dfrac{2}{5}
\\\\&=
\dfrac{1}{8}\cdot\dfrac{5}{5}-\dfrac{2}{5}\cdot\dfrac{8}{8}
&\text{(change to similar fractions)}
\\\\&=
\dfrac{5}{40}-\dfrac{16}{40}
\\\\&=
\dfrac{-11}{40}
\\\\&=
-\dfrac{11}{40}
.\end{align*}
