- #1
fishshoe
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Homework Statement
Prove: If a matrix A commutes with all matrices B \in M_{nxn}(F), then A must be scalar - i.e., A=diag.(λ,...,λ), for some λ \in F.
Homework Equations
If two nxn matrices A and B commute, then AB=BA.
The Attempt at a Solution
I understand that if A is scalar, it will definitely commute with all nxn matrices. But I don't get the intuition behind why commuting with more than one matrix implies that A must be scalar. The way I tried to solve it was by comparing an individual entry in the product, (AB)_{ij} = (BA)_{ij} = (AC)_{ij} = (CA)_{ji}, etc. This implies that
Ʃa_{ik}b_{kj} = Ʃb_{ik}a_{kj} = ...
But I'm not sure how that implies that A is scalar.