# Which of the subsets of PSKR^{3×3}ZSK are subspaces of PSKR^{3×3}ZSK? NSK a)

Which of the subsets of ${R}^{3×3}$ are subspaces of ${R}^{3×3}$?
a) The invertible $3×3$ matrices
b) The $3×3$ matrices whose entries are all integers
c) The $3×3$ matrices with all zeros in the third row
d) The non-invertible $3×3$ matrices
e) The diagonal $3×3$ matrices
f) The symmetric $3×3$ matrices
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Steve Hirano
encolatgehu
If the entries are integers and the linear combination includes $\pi$ as a coefficient, then the entries in the resulting matrix will usually not be integers. Just try it in some simple cases where there are just two terms, with coefficients $\pi$ and 0.
As for (c), a sum of non-invertible matrices is often invertible.
(e) and (f) are subspaces.
It is redundant to say they include 0 if they're closed under linear combinations, because there is a linear combination in which all coefficients are zero.