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Matrix Multiplication Associative Property: Key Concepts

Linear Algebra: Master Core Concepts & Applications

Linear Equations with Matrices

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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

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

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Intro

Ex.1

Ex.2

Ex.3

In this section, we will learn about the properties of matrix to matrix multiplication. These properties include the associative property, distributive property, zero and identity matrix property, and the dimension property. You will notice that the commutative property fails for matrix to matrix multiplication. Lastly, you will also learn that multiplying a matrix with another matrix is not always defined. The product of the two matrices is only defined if the number of columns in the first matrix is equal to the number of rows in the second matrix.