Answer
The student multiplied by the determinant instead of dividing by the determinant.
Work Step by Step
We know the following equation for the inverse of a matrix:
$$\begin{pmatrix}a\:&\:b\:\\ c\:&\:d\:\end{pmatrix}^{-1}=\frac{1}{\det \begin{pmatrix}a\:&\:b\:\\ c\:&\:d\:\end{pmatrix}}\begin{pmatrix}d\:&\:-b\:\\ -c\:&\:a\:\end{pmatrix}$$
Thus:
$$\frac{1}{\det \begin{pmatrix}2&4\\ 1&5\end{pmatrix}}\begin{pmatrix}5&-4\\ -1&2\end{pmatrix} \\ \begin{pmatrix}\frac{5}{6}&-\frac{2}{3}\\ -\frac{1}{6}&\frac{1}{3}\end{pmatrix}$$
