What does it mean for a matrix to be singular?

A singular matrix is one which has a determinant of zero. This has several important consequences depending on the context in which the matrix is being used: Firstly, it implies that the matrix is not invertible (this can be seen because we know that det(AB) = det(A)det(B), so that det(A inv(A)) = 1). Secondly, it means that, if the matrix describes a system of linear equations to be solved, then the solution is not unique, and not even guaranteed to exist. Finally, it tells us that the matrix must have at least one eigenvalue who value is zero.