I know the normal process is to do row operations to transform the matrix to get the identity matrix and then apply the same row operations in the identity matrix to get the inverse. But this process seems so tedious and I'd rather not waste so much time and energy doing the row operations to get it.
Is there some sort of general formula or shortcut to obtain the inverse similar to how the inverse of a 2x2's inverse is found? Thanks.
One way is to recall that, for $\operatorname{det}(A)\neq 0$,
$$A^{-1} = \frac{1}{\operatorname{det}(A)} \operatorname{adj}(A)$$
and use the Rule of Sarrus to calculate $\operatorname{det}(A)$. Whether that is easier or not is debatable, but the formula for a $2\times 2$ matrix is basically this.