Help with a simple matrix proof

  • Thread starter Thread starter paulrb
  • Start date Start date
  • Tags Tags
    Matrix Proof
Click For Summary
SUMMARY

The discussion focuses on proving that the product of two square matrices, AB, commutes with a third square matrix C, given that both A and B commute with C. The key equations utilized are AC = CA and BC = CB, which establish the commuting relationships. The associative property of matrix multiplication is crucial in manipulating the expression (AB)C to demonstrate that it equals C(AB). This proof is straightforward once the properties of matrix multiplication are applied correctly.

PREREQUISITES
  • Understanding of matrix multiplication
  • Familiarity with square matrices
  • Knowledge of the associative property of multiplication
  • Basic concepts of commuting matrices
NEXT STEPS
  • Study the properties of matrix multiplication in detail
  • Learn about commuting matrices and their implications in linear algebra
  • Explore proofs involving matrix equations and identities
  • Investigate applications of matrix commutativity in various mathematical contexts
USEFUL FOR

Students studying linear algebra, mathematicians interested in matrix theory, and educators teaching concepts related to matrix operations and properties.

paulrb
Messages
18
Reaction score
1
Edit: Sorry, I figured this out shortly after posting. It's a simple problem but I hadn't used matrix equations before.

Homework Statement



If A, B, and C are all square matrices of the same size, show that AB commutes with C if A and B both commute with C.

Homework Equations



Formula for matrix multiplication

The Attempt at a Solution



All I know is
AC = CA
BC = CB
A, B, and C are square matrices of the same size

I want to prove that (AB)(C) = (C)(AB).
But I am not sure how to do this, or even how to get started.
 
Physics news on Phys.org
Well Matrix multiplication is associative so you should be able to manipulate (AB)C to get C(AB) pretty easily.
 

Similar threads

Hermitian Matrix and Commutation relations
  • · Replies 2 ·
Replies
2
Views
3K
Does there exist a 2x2 non-singular matrix with only one 1d eigenspace?
  • · Replies 2 ·
Replies
2
Views
2K
Linear Algebra, Find a matrix C st CA = B
  • · Replies 2 ·
Replies
2
Views
3K
Proof regarding determinant of block matrices
  • · Replies 5 ·
Replies
5
Views
5K
Prove the Extended Law of Sines
  • · Replies 1 ·
Replies
1
Views
3K
All 2x2 Hermitian and Unitary Matrices (Check My Proof)
  • · Replies 3 ·
Replies
3
Views
4K
Normal matrix as sum of self adjoint and commuting matrices
  • · Replies 5 ·
Replies
5
Views
3K
MHB Does Commutativity Hold for Matrices A and B with a Specific Matrix C?
  • · Replies 3 ·
Replies
3
Views
2K
Proving facts about matrices without determinants
  • · Replies 20 ·
Replies
20
Views
3K
Understanding Matrices Sums and Products
  • · Replies 3 ·
Replies
3
Views
2K