Answer
$$A^{-1}=-\frac{36}{143}\left[ \begin {array}{cccccc} \frac{8}{9}&-\frac{9}{4}\\ -\frac{5}{3}&-\frac{1}{4}\end {array} \right].$$
Work Step by Step
Since $ad-bc=-\frac{1}{4}\frac{8}{9}-\frac{9}{4}\frac{5}{3}=-\frac{143}{36}\neq 0$, then $A$ has inverse given by $$A^{-1}=-\frac{36}{143}\left[ \begin {array}{cccccc} \frac{8}{9}&-\frac{9}{4}\\ -\frac{5}{3}&-\frac{1}{4}\end {array} \right].$$
