The so-called PAQ problem is concerned with the solution of sparse systems of linear equations $Ax=b$ using the transformation $PAQy=Pb, x=Qy$. An algorithm is given for choosing $P$ and $Q$ to partition the matrix $A$ into its irreducible components. A theorem on which this algorithm is based has long been known, yet no simple, easily understood proof appears in the literature. Such a proof is given here. Remarks are made concerning some unsolved problems related to the PAQ problem.