Contents

- 1 Can matrix multiplication be commutative?
- 2 What makes a matrix commutative?
- 3 Is multiplication always commutative?
- 4 Is matrix multiplication Abelian group?
- 5 Is matrix multiplication reversible?
- 6 Where is matrix multiplication used?
- 7 Is the rref of a matrix unique?
- 8 Is diagonal matrix multiplication commutative?
- 9 Is commutative property of subtraction?
- 10 How do you explain commutative property of multiplication?
- 11 What is the key word for commutative property?
- 12 Is matrix multiplication associative?
- 13 Are matrices a group?
- 14 How do you know if a group is Abelian?

## 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.

## 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 **word** “**commutative**” 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.

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

**Ways to Show a Group is Abelian**

- 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.
- Show the
**group**is isomorphic to a direct product of two**abelian**(sub)**groups**. **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.