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Which of the following subsets of $\mathbb{R}^{3 \times 3}$ are subspaces of $\mathbb{R}^{3 \times 3}$?

A. The $3 \times 3$ matrices with determinant 0
B. The $3 \times 3$ matrices whose entries are all integers
C. The invertible $3 \times 3$ matrices
D. The $3 \times 3$ matrices with all zeros in the first row
E. The diagonal $3 \times 3$ matrices
F. The symmetric $3 \times 3$ matrices

I answered B, D, E and F, but it appears to be incorrect. How so?

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(B) is false, since there are $\;3\times3\;$ integer matrices which multiplied by $\;\frac12\;$ aren't integer anymore (example?)

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