Gapplet

The square matrices with rational entries of a given dimension form a ring. Let A be a square matrix over Q.

A is invertible if and only if its determinant is nonzero.

By the theorem, if A is invertible, it is not a zero divisor. If A is not invertible, we show it is a zero divisor by giving a matrix B such that BA = 0. Of course, the matrix B has to do with the kernel of the matrix A.