 # Which of the subsets of PSKR^{3×3}ZSK are subspaces of PSKR^{3×3}ZSK? NSK a) Inyalan0 2021-12-11 Answered
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
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it 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.