"2 times row one of $\displaystyle A^{-1}$ minus row two of $\displaystyle A^{-1}$ plus 3 times row 3 of $\displaystyle A^{-1}$" is simply $\displaystyle \begin{bmatrix}2 & -1 & 3\end{bmatrix}A^{-1}$. But that is the second row of A. Multiplying the second row of A by $\displaystyle A^{-1}$ gives the second row of the identity matrix which is $\displaystyle \begin{bmatrix}0 & 1 & 0\end{bmatrix}$.