Proving the scalar matrices are the center of the matrix ring

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SUMMARY

Scalar matrices are definitively established as the center of the ring of matrices, denoted as M(R), where R is a commutative ring with identity. The proof involves demonstrating that any matrix C in the center must be diagonal; if C is not diagonal, it fails to commute with certain matrices, leading to a contradiction. Furthermore, if C is diagonal but has distinct diagonal entries, it also fails to commute with specific matrices in M(R). Thus, scalar matrices are the only matrices that commute with all others in M(R).

PREREQUISITES
  • Understanding of matrix algebra and properties of matrix rings.
  • Familiarity with scalar matrices and their definitions.
  • Knowledge of commutative rings and their identities.
  • Basic proof techniques in linear algebra.
NEXT STEPS
  • Study the properties of scalar matrices in linear algebra.
  • Learn about the structure of matrix rings, specifically M(R).
  • Explore proof techniques for commutativity in matrix operations.
  • Investigate examples of non-scalar matrices and their commutation properties.
USEFUL FOR

This discussion is beneficial for mathematicians, students of linear algebra, and anyone interested in the properties of matrix rings and scalar matrices.

Bipolarity
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I read that scalar matrices are the center of the ring of matrices. How would I prove this?
Tips are appreciated. It is already obvious that scalar matrices commute with all matrices, but the converse seems tricky.

BiP
 
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Supposing R is commutative with identity:

Let C be in the center of M(R). If C is not diagonal then cij≠0 for some i≠j. Letting A be the matrix with 1 in the ji position and 0 elsewhere, we see that CA≠AC, contradicting the fact that C is in the center of M(R). So C is diagonal. If C is diagonal but cii≠cjj, then we can again find a matrix in M(R) which does not commute with C.
 

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