For the matrix of a complex number z=a+bi, we note that△=a2+b2, which is never zero except when z=0–but of course the number 0 never had a reciprocal before, and that remains the case in the wider arena of the complex numbers. This does conrm however that every non-zero complex number possesses a multiplicative inverse.
We stand here on the edge of the vast worlds of linear algebra,representation theory, and applications to multi-dimensional calculus, and this is not the place to go further. However, the reader should be aware that matrices apply to three dimensions and indeed to n-dimensional space, typically through n×n matrices. Although the arrays become larger and more complicated, the matrices themselves yet remain two-dimensional numerical objects.