The group axioms demand that one has an inverse element. When it comes to multiplication zero has no inverse element because one cannot reconstruct the original number after it was multiplied by zero:
There is an irreversible loss of information!
Therefore all numbers cannot form a group under multiplication as long as zero is included.
When it comes to matrices some matrices have an inverse, some don't have one. Singular matrices have determinant zero and don't have one. When the column vectors are linear dependent the determinant is zero and the matrix is singular.
Where does the loss of information happen here? How is 'more' information lost when a vector is multiplied by a singular matrix (compared to multiplying it by an invertible matrix)?