Matrix Multiplication Properties for 2x2 Matrices

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SUMMARY

The discussion centers on the properties of matrix multiplication specifically for 2x2 matrices. It establishes that while matrix multiplication is generally non-commutative, the equation (A^n)(A^m) = (A^m)(A^n) holds true due to the associative property of multiplication. Both sides of the equation contain the same number of matrix multiplications, n+m, just arranged differently. This confirms the associative nature of matrix multiplication for 2x2 matrices.

PREREQUISITES
  • Understanding of matrix operations, specifically 2x2 matrices
  • Familiarity with the concepts of commutativity and associativity in mathematics
  • Basic knowledge of exponentiation in the context of matrices
  • Ability to manipulate and simplify matrix expressions
NEXT STEPS
  • Study the properties of matrix multiplication in greater depth
  • Explore the implications of non-commutativity in larger matrices
  • Learn about the applications of matrix exponentiation in linear algebra
  • Investigate other matrix properties such as determinants and eigenvalues
USEFUL FOR

Students of linear algebra, mathematicians, and anyone interested in understanding the properties of matrix operations and their implications in various mathematical contexts.

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Where A = a 2*2 matrix, is the following true:

(A^n)(A^m) = (A^m)(A^n)

Thanks in advance
 
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Take a guess. Tell us why it might be true.
 
Well I think it would be true, however I know that matrix multiplication is non-commutative so I wasnt sure.

Thanks
 
Matrix multiplication isn't commutative in general. But this is special. It IS associative. Both sides of that equation have n+m A's. They are just grouped differently.
 

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