Commutative Property

Examples of Commutative Property in the following topics:

• Basic Operations

  • The commutative property describes equations in which the order of the numbers involved does not affect the result.
  • Addition and multiplication are commutative operations:
  • The associative property describes equations in which the grouping of the numbers involved does not affect the result.
  • As with the commutative property, addition and multiplication are associative operations:
  • The distributive property can be used when the sum of two quantities is then multiplied by a third quantity.

• Adding and Subtracting Polynomials

  • When adding polynomials, the commutative property allows us to rearrange the terms to group like terms together.

• Multiplying Polynomials

  • Multiplying a polynomial by a monomial is a direct application of the distributive and associative properties.
  • Recall that the distributive property says that
  • for all real numbers $a,b$ and $c.$ The associative property says that
  • For convenience, we will use the commutative property of addition to write the expression so that we start with the terms containing $M_1(x)$ and end with the terms containing $M_n(x)$.
  • Explain how to multiply polynomials using the distributive property and describe the results of doing so

• Adding and Subtracting Algebraic Expressions

  • The Commutative Property of Addition says that we can change the order of the terms without changing the sum.

• The Identity Matrix

  • The matrix that has this property is referred to as the identity matrix.
  • The identity matrix, designated as $[I]$, is defined by the property: $[A][I]=[I][A]=[A]$.
  • This stipulation is important because, for most matrices, multiplication does not commute.
  • What matrix has this property?
  • There is no identity for a non-square matrix, because of the requirement of matrices being commutative.

• Negative Numbers

  • The basic properties of addition (commutative, associative, and distributive) also apply to negative numbers.
  • For example, the following equation demonstrates the distributive property:

• Introduction to Variables

  • Variables allow one to describe some mathematical properties.
  • For example, a basic property of addition is commutativity, which states that the order of numbers being added together does not matter.
  • Commutativity is stated algebraically as $\displaystyle (a+b)=(b+a)$.

• Addition, Subtraction, and Multiplication

  • The preceding definition of multiplication of general complex numbers follows naturally from this fundamental property of the imaginary unit.
  • = $ac + bidi + bci + adi$ (by the commutative law of addition)
  • = $ac + bdi^2 + (bc + ad)i$ (by the commutative law of multiplication)
  • = $(ac - bd) + (bc + ad)i$ (by the fundamental property of the imaginary unit)

• Addition and Subtraction; Scalar Multiplication

  • Matrix addition is commutative and is also associative.  
  • Scalar multiplication has the following properties:

• The Inverse of a Matrix

  • The inverse of matrix $[A]$ is $[A]^{-1}$, and is defined by the property: $[A][A]^{-1}=[A]^{-1}[A]=[I]$.
  • The inverse of matrix $[A]$, designated as $[A]^{-1}$, is defined by the property: $[A][A]^{-1}=[A]^{-1}[A]=[I]$, where $[I]$ is the identity matrix.
  • Note that, just as in the definition of the identity matrix, this definition requires commutativity—the multiplication must work the same in either order.
  • Practice finding the inverse of a matrix and describe its properties