If the product abcd=0, then we know at least one of a, b, c, d is zero.
Therefore the set would include the zero matrix, in which a=b=c=d=0, as well as all combinations of a, b, c, d where at least one is zero.
ADDITION
The implication is that V is not closed under addition because the result of adding two matrices in V is:
( a1 b1 ) + (a2 b2 ) = ( a1+a2 b1+b2 )
( c1 d1 ) ( c2 d2 ) ( c1+c2 d1+d2 )
Unless corresponding elements in the matrices on the left produce at least one zero in an element on the right, the sum will not belong to V. So a1+a2=0, making a2=-a1 or b1+b2=0, etc. So in the general case V is not closed under addition. If, for example, a1=0=b2 but no other element is zero, the sum will contain no zero elements.
MULTIPLICATION
Matrix multiplication gives us:
( a1a2+b1c2 a1b2+b1d2 )
( c1a2+d1c2 c1b2+d1d2 )
Again, at least one element in the matrix product must be zero in order to qualify for membership of V. Therefore V is not closed under matrix multiplication. V is closed under scalar multiplication, because zero elements remain zero.
My guess is that V is not a vector subspace since matrix addition and matrix multiplication are excluded in general between the elements of the set. The identity vector, like the zero vector, appears to be an element in the set because the matrix contains two zero elements.