## What is commutative property?

The commutative property is a property of mathematical operations that states that the order in which the operands are applied does not affect the result of the operation. The following are examples of commutative property:

**Addition of real numbers**: For any two real numbers a and b, a + b = b + a.

For example, 2 + 3 = 3 + 2 = 5.

**Multiplication of real numbers**: For any two real numbers a and b, a x b = b x a.

For example, 2 x 3 = 3 x 2 = 6.

**Addition of matrices**: For any two matrices A and B with the same dimensions, A + B = B + A.

For example, if A = [1 2; 3 4] and B = [5 6; 7 8], then A + B = [1+5 2+6; 3+7 4+8] = [6 8; 10 12], and B + A = [5+1 6+2; 7+3 8+4] = [6 8; 10 12].

**Multiplication of complex numbers**: For any two complex numbers z1 and z2, z1 x z2 = z2 x z1.

For example, (2 + 3i) x (4 – i) = (4 – i) x (2 + 3i) = 11 + 10i.

**Intersection of sets**: For any two sets A and B, A âˆ© B = B âˆ© A.

For example, if A = {1, 2, 3} and B = {2, 3, 4}, then A âˆ© B = {2, 3} and B âˆ© A = {2, 3}.

## The commutative property for matrix addition

The commutative property is one of the fundamental properties of matrix addition. It states that the order of matrices can be changed without affecting the result of the addition operation. In other words, for any two matrices A and B, A + B = B + A.

Mathematically, the commutative property of matrix addition can be expressed as follows:

For any two matrices A and B, the following equation holds: A + B = B + A

This property is similar to the commutative property of the addition of real numbers. For example, the commutative property of addition of real numbers states that for any two real numbers a and b, a + b = b + a.

The commutative property of matrix addition can be proven using the definition of matrix addition. Recall that to add two matrices A and B, we add the corresponding entries of A and B to get the corresponding entries of the sum matrix C. That is, if A and B have the same dimensions, then we can define their sum C as follows:

C[i,j] = A[i,j] + B[i,j]

where C[i,j] represents the (i,j)-th entry of the sum matrix C.

Using this definition, we can prove the commutative property as follows:

A + B = C, where C[i,j] = A[i,j] + B[i,j] for all i,j.

B + A = D, where D[i,j] = B[i,j] + A[i,j] for all i,j.

By the definition of matrix addition, we have:

C[i,j] = A[i,j] + B[i,j] = B[i,j] + A[i,j] = D[i,j]

Therefore, A + B = B + A, which proves the commutative property of matrix addition.

## Basic questions on commutative property

Q1. State the commutative property of addition for real numbers.

Solution: The commutative property of addition for real numbers states that for any two real numbers a and b, a + b = b + a. That is, the order in which we add two real numbers does not affect their sum.

Q2. Give an example of the commutative property of multiplication for complex numbers.

Solution: The commutative property of multiplication for complex numbers states that for any two complex numbers z1 and z2, z1 x z2 = z2 x z1. For example, (2 + 3i) x (4 – i) = 5 + 10i, and (4 – i) x (2 + 3i) = 5 + 10i.

Q3. Is subtraction commutative? Explain your answer.

Solution: No, subtraction is not commutative. That is, for any two real numbers a and b, a – b is not equal to b – a. For example, 5 – 3 = 2, but 3 – 5 = -2.

Q4. For any two matrices A and B, is (A + B) – A = B? Why or why not?

Solution: No, (A + B) – A is not equal to B in general. Simplifying the left-hand side, we get:

(A + B) – A = A – A + B = 0 + B = B.

So, (A + B) – A = B if and only if A – A = 0, which is true only if A is the zero matrices.

Q5. Give an example of two sets that have the commutative property for the intersection.

Solution: The commutative property of intersection states that for any two sets A and B, A âˆ© B = B âˆ© A. For example, if A = {1, 2, 3} and B = {2, 3, 4}, then A âˆ© B = {2, 3} and B âˆ© A = {2, 3}.

Q6. Is the commutative property true for the division of real numbers? Why or why not?

Solution: No, the commutative property is not true for the division of real numbers. That is, for any two real numbers a and b, a Ã· b is not equal to b Ã· a in general.

Q7. Does the commutative property hold for the operation of matrix multiplication? Explain your answer.

Solution: No, the commutative property does not hold for matrix multiplication. That is, for any two matrices A and B, A x B is not equal to B x A in general. In fact, the order of multiplication matters in matrix multiplication, and reversing the order of matrices can result in a different product.

Q8. Give an example of two matrices that do not satisfy the commutative property of addition.

Solution: Let A = [1 2; 3 4] and B = [5 6; 7 8]. Then, A + B = [6 8; 10 12] and B + A = [6 8; 10 12]. So, A + B = B + A, and the commutative property of addition holds for these matrices. However, let C = [1 2; 3 4] and D = [5 6; 7 8; 9 10]. Then, C + D is not defined, because the matrices have different dimensions. So, the commutative property of addition does not make sense