The adjoint $A\adj$ of a matrix $A$ is the transpose of the matrix of the cofactors of the elements of $A$. The computation of the adjoint from its definition involves the computation of $n^{2}$ determinants of order $(n-1)$,---,a prohibitively expensive $O(n^{4})$ process. On the other had the computation from the formula $A\adj = \det(A)A\inv$ breaks down when $A$ is singular and is potentially unstable when $A$ is ill-conditioned. In this paper we first show that the ajdoint can be perfectly conditioned, even when $A$ is ill-conditioned. We then show that if due care is taken the adjoint can be accurately computed from the inverse, even when the latter has been inaccurately computed. In an appendix to this paper we establish a folk result on the accuracy of computed inverses. \end{minipage} \end{center} Also cross-referenced as UMIACS-TR-98-02