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Matrix Addition Properties: Commutative and Associative Rules

Linear Algebra: Master Core Concepts & Applications

Linear Equations with Matrices

Row reduction and echelon forms

Linear combination and vector equations in \(R^n\)

Matrix equation Ax=b

Solution sets of linear systems

Application of linear systems

Solving systems of linear equations by graphing

Notation of matrices

The three types of matrix row operations

Representing linear system as a matrix

Representing a linear system as a matrix

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Solving a linear system with matrices using Gaussian elimination

Linear Transformation

Linear independence

Image and range of linear transformations

Properties of linear transformation

The matrix of a linear transformation

Matrix Operations

Adding and subtracting matrices

Multiplying a matrix by a scalar

Multiplying a matrix by another matrix

Matrix multiplication

Zero matrix

Identity matrix

Properties of matrix addition

Properties of matrix scalar multiplication

Properties of matrix to matrix multiplication

Properties of matrix multiplication

Inverse of Matrices

The invertible matrix theorem

The inverse of a 2 x 2 matrix

The inverse of a 3 x 3 matrices with matrix row operations

2 x 2 invertible matrix

Solving linear systems using 2 x 2 inverse matrices

Subspace of \(\Bbb{R}^n\)

Properties of subspace

Column space

Null space

Dimension and rank

Determinant of a Matrix

Properties of determinants

The determinant of a 2 x 2 matrix

The determinant of a 3 x 3 matrix (General & Shortcut Method)

Solving linear systems using Cramer's Rule

Solving linear systems using Cramer"s Rule

Cramer"s Rule with 2 x 2 matrices

Eigenvalue and Eigenvectors

Eigenvalues and eigenvectors

The characteristic equation

Diagonalization

Complex eigenvalues

Discrete dynamical systems

Orthogonality and Least Squares

Inner product, length, and orthogonality

Orthogonal sets

Orthogonal projections

Least-squares problem

Applications to linear models

Parent Assignments

Challenge Zone

Foundation Review

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Intro

Overview of five key properties of matrix addition

Ex.1

Ex.2

Ex.3

Verifying the zero matrix property with a 3 by 3 matrix

Ex.4

Proving the identity matrix times X equals X for a 2x2 matrix

Verifying the commutative property of matrix addition

Verifying the associative property of matrix addition

Proving the dimension property by adding three 2x2 matrices

Commutative property of addition with 3x3 identity and zero matrices

Proving the associative property with identity and zero matrices

Verifying the dimension property by adding three 3x3 matrices

In this section, we will look at the properties of matrix addition. These properties include the dimension property for addition, commutative property, and associative property. Note that for the dimension property, we are allowed to add or subtract two matrices with the same dimensions. The commutative property states that changing the order of the addition or subtraction of two matrices lead to the same result. For the associative property, changing what matrices you add or subtract one will lead to the same answer. There are also matrix addition properties for identity and <a href="[[glossary.html|{"keyword":"Zero matrix"}]]">zero matrices</a> as well. Adding a zero matrix with another matrix (call it A) will give back A. Multiplying an <a href='[[glossary.html|{"keyword":"Identity matrix"}]]'>identity matrix</a> with another matrix (call it B) will give back B.

Rules for matrix addition and practice questions

Mastering Matrix Addition Properties: Commutative and Beyond

What You'll Learn

Identify the dimension property requiring equal dimensions for matrix addition

Apply the commutative property to show matrix addition order does not change results

Verify the associative property for grouping matrices in addition

Recognize zero matrix property when adding zero matrices returns the original matrix

Understand identity matrix property where multiplying by identity preserves the matrix

What You'll Practice

Verifying commutative property by computing A+B and B+A

Confirming associative property with three-matrix expressions

Adding matrices to demonstrate dimension property holds

Showing zero matrix property with various matrix dimensions

Why This Matters

Before You Start — Make Sure You Can:

This Unit Includes

9 Video lessons

Practice exercises

Learning resources

Skills

Matrix Addition

Commutative Property

Associative Property

Dimension Property

Zero Matrix

Identity Matrix

Linear Algebra