A 2-by-2 matrix is not invertible if its determinant=0.
So:
| a b |
| c d | =ad-bc≠0⇒matrix is invertible.
If we take the sum of two invertible matrices X and Y:
⎛ x11 x12 ⎞ ⎛ y11 y12 ⎞
⎝ x21 x22 ⎠ ⎝ y21 y22 ⎠=
⎛ x11+y11 x12+y12 ⎞
⎝ x21+y21 x22+y22 ⎠
This sum must also belong to W for the set to be closed. However, if it can be shown that this is not necessarily an invertible matrix, then the sum doesn't belong to W.
An example of a non-invertible M2×2 is Z=
⎛ 3 4 ⎞
⎝ 9 12 ⎠because its determinant is 3×12-4×9=0
If this matrix can be the sum of 2 invertible matrices then we've found that under matrix addition we can generate a matrix which doesn't belong to W.
Let X and Y be:
⎛ 1 2 ⎞ ⎛ 2 2 ⎞
⎝ 3 4 ⎠ ⎝ 6 8 ⎠
So X+Y=Z and, since X and Y are invertible, because their determinants are non-zero, their sum is Z∉W. So W cannot be a subspace of V (M2×2).
(Incidentally, X-1=
⎛ -2 1 ⎞
⎝ 3/2 -½ ⎠and
Y-1=
⎛ 2 -½ ⎞
⎝ -3/2 ½ ⎠so they are invertible.)