Its -1/2 not the one you put on the matrix

The matrix $A = \begin{pmatrix} 5 & -2 \\ 3 & -1 \end{pmatrix}$ was taken directly from the image you provided. There is no $-1/2$ in this matrix, nor does it appear in the calculation of its inverse or the invariant lines.

For part a), my analysis showed that the lines $y=2x$ and $y=\frac{3}{4}x$ are not invariant under the transformation $T$ represented by matrix $A$, as points on these lines are not mapped back onto the same lines (except for the origin in the case of $y=2x$).

For part b), the calculation of the inverse matrix $A^{-1}$ is:

$$A^{-1} = \frac{1}{(5)(-1) - (-2)(3)} \begin{pmatrix} -1 & 2 \\ -3 & 5 \end{pmatrix} = \frac{1}{-5+6} \begin{pmatrix} -1 & 2 \\ -3 & 5 \end{pmatrix} = \begin{pmatrix} -1 & 2 \\ -3 & 5 \end{pmatrix}$$

This calculation does not involve $-1/2$.

Could you please clarify which part you are referring to?