A method is given for finding the reciprocal of a matrix which is triply partitioned horizontally and vertically in such a manner that the sub-matrices in the principal diagonal are square, but these matrices need not be of the same order. A preliminary rearrangement of the matrix may be helpful.

In theoretical work it is sometimes required to find the reciprocal of a matrix which can be so partitioned that some of the sub-matrices are of simple types, e.g. triangular or nul matrices, and the calculation of the reciprocal may then be facilitated by using formulae for the inversion of partitioned matrices. The same method will be often advantageous in numerical work also. In an earlier paper the reciprocation of doubly-partitioned matrices was treated in a general way and a method was given for triply-partitioned matrices subject to the restriction that the sub-matrices in the principal diagonal are square and of the same order.