Let A be a fixed vector in R^(n*n) and let S be the set of all matrices that commute with A; that is,
S = { B | AB = BA }
Show that S is a subspace of R^(n*n)
The work I did so far:
if B in S then AB = BA => A (alpha B) = Alpha (AB) = Alpha (BA) => Alpha B is in S.
Same closure under addition holds:
if B and C are matrices in S then AB=BA and AC=CA => AB + AC = BA+CA => A(B+C) = (B+C)A => B + C in S.
However, I don't know how to show that the set S is not empty, and i'm also not sure about what I did above

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I saw this question earlier and did the exact same proofs for scalar mult and addition closure

but what about showing that S is not empty ?

what about \(I\) that is a member, no?

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