A singular matrix is one that cannot be inverted. This means that solution techniques that use division by any such matrix cannot be employed.
Here, “dividing” refers to the notion that because we have (for example) Distance = Rate * Time, we can calculate rate by inverting the equation: Time = Distance / Rate. If “Rate” is singular, it cannot be inverted…so we cannot actually execute the analysis (even though the theory would hold). Of course, things can get much more complicated than this simple example!