Is the Matrix subset a Subspace?

Question

If 2×2 is the vector space of 2×2 real matrices. Let be the set of all 2×2 matrices whose two columns are orthogonal to one another.

I believe it's(W) no it is not a subspace for the 2 following criteria:

1. I believe the all 4 zeroes matrix fails because it's not orthogonal. Although I did want to confirm that the zero vector cannon be orthogonal to itself.

2. The "closed under addition criteria": let A = [[a1,a2],[a3,a4]] and B = [[b1,b2],[b3,b4]]

let: a1a2 + a3a4 = 0 and b1b2 + b3b4 = 0; ... because of dot product orthogonality.

but A+B = [[(a1+b1),(a2+b2)],[(a3+b3),(a4+b4)]] this yields: (a1+b1)(a2+b2) +(a3+b3)(a4+b4) =

a1a2 + b1b2 + a1b2+ a2b1 + a3a4 + b3b4 + a3b4 +a4b3

The bold ones could be anything even if the non bold yield zero; so via this proof it fails to be orthogonal.

Are my two proofs correct? And if not then is W a subspace?