Matrices: Abelian Group?

The set of matrices is an abelian group?

To determine whether a set of matrices forms an abelian group, we need to be more specific — for example:

1. Which set of matrices?

   • All square matrices?
   • Invertible matrices?
   • Diagonal matrices?
   • Symmetric matrices?
   • Upper/lower triangular matrices?
   • Over which field? (Real numbers ℝ, complex numbers ℂ, etc.)

2. Which operation?

   • Matrix addition or matrix multiplication?

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Let's go through common cases:

✅ Under matrix addition:

• The set of all n × n matrices over a field (like ℝ or ℂ) is an abelian group under addition:

  • Closure: Sum of two n × n matrices is an n × n matrix.
  • Associativity: Matrix addition is associative.
  • Identity: The zero matrix.
  • Inverses: Each matrix has an additive inverse (its negative).
  • Commutativity: Matrix addition is commutative.

✅ So yes, (ℝⁿˣⁿ, +) is an abelian group.

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❌ Under matrix multiplication:

• The set of all n × n matrices over a field is not a group under multiplication:

  • Not all matrices are invertible.
  • Matrix multiplication is not commutative in general.

❌ So it is not even a group, let alone an abelian one.

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✅ Invertible matrices under multiplication:

• The set of invertible n × n matrices (called GL(n, ℝ) or GL(n, ℂ)) forms a group under multiplication.

  • But it is not abelian for n > 1 (since multiplication is not commutative).

✅ Group? Yes ❌ Abelian? No (for n > 1)

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Summary:

Set of Matrices	Operation	Group?	Abelian?
All n × n matrices (ℝⁿˣⁿ)	Addition	Yes	Yes
All n × n matrices (ℝⁿˣⁿ)	Multiplication	No	No
Invertible n × n matrices (GL(n))	Multiplication	Yes	No
1 × 1 invertible matrices	Multiplication	Yes	Yes

Let me know which specific set and operation you're referring to if you want a more targeted answer.