Use one of the stated properties of eigenvalues to prove that a matrix is singular if and only if it has 0 as one of its eigenvalues.

A matrix is singular if and only if its determinant is 0. The product of the eigenvalues of a matrix equals its determinant. A product of numbers equals 0 if and only if one of those numbers is 0. Accordingly, a matrix is singular if and only if one of its eigenvalues is 0.