A problem of Cayley from 1857 and how he could have solved it

In a paper of 1857 Cayley suggests the problem of determining all the symmetric functions of the common solutions of a linear/cubic homogeneous system of ternary equations, as functions of the coefficients of the equations. We show how Cayley could have solved the problem himself if he had taken another look at it. All the facts necessary for the solution were known to Cayley, including a crucial formula known as Cayley's Identity. We study the problem by mixing techniques from vector symmetric function theory with techniques from the study of determinants and permanents. We develop some new formulas for determinants of matrices which arise by the Hadamard or pointwise product from other matrices.