Condensation of Determinants

By the Rev. C. L. Dodgson, M.A., Student of Christ Church, Oxford.

Communicated by the Rev. Bartholomew Price, M.A., F.R.S. Received May 15, 1866.

If it be proposed to solve a set of simultaneous linear equations, not being all homogeneous, involving unknowns, or to test their compatibility when all are homogeneous, by the method of determinants, in these, as well as in other cases of common occurrence, it is necessary to compute the arithmetical values of one or more determinants—such, for example, as

Now the only method, so far as I am aware, that has been hitherto employed for such a purpose, is that of multiplying each term of the first row or column by the determinant of its complemental minor, and affecting the products with the signs and alternately, the determinants required in the process being, in their turn, broken up in the same manner until determinants are finally arrived at sufficiently small for mental computation.

This process, in the above instance, would run thus:—

But such a process, when the block consists of , , or more terms, is so tedious that the old method of elimination is much to be preferred for solving simultaneous equations; so that the new method, excepting for equations containing or unknowns, is practically useless.

The new method of computation, which I now proceed to explain, and for which “Condensation” appears to be an appropriate name, will be found, I believe, to be far shorter and simpler than any hitherto employed.