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Master Matrix Multiplication: Rules, Examples, and Applications

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 lesson, we will learn how to multiply a matrix with another matrix. But we will learn about n-tuples first. An n-tuple is an ordered list of n numbers. Multiplying an n-tuple by another n-tuple is called the dot product. The dot product is the summation of all product of each corresponding entries. To multiply a matrix with another matrix, we have to think of each row and column as a n-tuple. Each entry will be the dot product of the corresponding row of the first matrix and corresponding column of the second matrix.  For example, if your entry is at the 3rd row and 4th column, then you have to take the dot product of the 3rd row of the first matrix and 4th column of the second matrix. Note that not all matrices can be multiplied.

How to multiply two matrices together