# FAQ: When is matrix multiplication commutative?

## Can matrix multiplication be commutative?

Matrix multiplication is not commutative

In other words, in matrix multiplication, the order in which two matrices are multiplied matters!

## What makes a matrix commutative?

If the product of two symmetric matrices is symmetric, then they must commute. Circulant matrices commute. They form a commutative ring since the sum of two circulant matrices is circulant.

## Is multiplication always commutative?

(Addition in a ring is always commutative.) In a field both addition and multiplication are commutative.

## Is matrix multiplication Abelian group?

The set Mn(R) of all n × n real matrices with addition is an abelian group. However, Mn(R) with matrix multiplication is NOT a group (e.g. the zero matrix has no inverse).

## Is matrix multiplication reversible?

Yes! Matrices are members of non commutative ring theory. Non commutative ring theory deals specifically with rings that are non-commutative with respect to multiplication.

## Where is matrix multiplication used?

1 Matrix multiplication. Matrix multiplication is probably the most important matrix operation. It is used widely in such areas as network theory, solution of linear systems of equations, transformation of co-ordinate systems, and population modeling, to name but a very few.

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## Is the rref of a matrix unique?

Theorem: The reduced (row echelon) form of a matrix is unique. Now interpret these matrices as augmented matrices. The system for R has a unique solution r or is inconsistent, respectively. Similarly, the system for S has a unique solution s or is inconsistent, respectively.

## Is diagonal matrix multiplication commutative?

Multiplication of diagonal matrices is commutative: if A and B are diagonal, then C = AB = BA.

## Is commutative property of subtraction?

The commutative property states that the numbers on which we operate can be moved or swapped from their position without making any difference to the answer. The property holds for Addition and Multiplication, but not for subtraction and division.

## How do you explain commutative property of multiplication?

The commutative property of multiplication tells us that we can multiply a string of numbers in any order. Basically: 2 x 3 x 5 will create the same answer as 3 x 5 x 2, or 2 x 5 x 3, etc. Hope this helps.

## What is the key word for commutative property?

The wordcommutative” comes from “commute” or “move around”, so the Commutative Property is the one that refers to moving stuff around. For addition, the rule is “a + b = b + a”; in numbers, this means 2 + 3 = 3 + 2. For multiplication, the rule is “ab = ba”; in numbers, this means 2×3 = 3×2.

## Is matrix multiplication associative?

Sal shows that matrix multiplication is associative. Mathematically, this means that for any three matrices A, B, and C, (A*B)*C=A*(B*C).

## Are matrices a group?

A group in which the elements are square matrices, the group multiplication law is matrix multiplication, and the group inverse is simply the matrix inverse.

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## How do you know if a group is Abelian?

Ways to Show a Group is Abelian

1. Show the commutator [x,y]=xyx−1y−1 [ x, y ] = x y x − 1 y − 1 of two arbitary elements x,y∈G x, y ∈ G must be the identity.
2. Show the group is isomorphic to a direct product of two abelian (sub)groups.
3. Check if the group has order p2 for any prime p OR if the order is pq for primes p≤q p ≤ q with p∤q−1 p ∤ q − 1.