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Response Details:
A subspace is a non-empty set that is closed under addition and scalar multiplication. So, the usual way to test to see whether a set is a subspace is to see whether addition of arbitrary elements in the set remains in the set, and whether arbitrary scalar multplication keeps you in the set.
Non-singular matrices do not form a subspace, because scalar multiplication by 0 gives you the 0 matrix, which is singular.
The set of matrices you gave is OK, except for the a^2 term in the second row. If you add 2 such matrices, you get the sum of squares a_1^2 + a_2^2 in that place. But that entry needs to be the square of the term above it, which would be a_1 + a_2. Since this is not true in general, it fails to be a subspace.
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