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Identity Matrix: Definition, Properties, and Applications

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

Ex.1

Ex.2

Intro to identity matrices and multiplicative inverses

Multiplying a 3x3 identity matrix by matrix A

Computing 2A + 4I₃ using scalar multiplication and matrix addition

Computing -4B + 2I₂ using scalar multiplication and matrix addition

Multiplying the 2x2 identity matrix I2 by matrix B

Multiplying a 4x4 identity matrix by the scalar zero

Verifying two matrices are multiplicative inverses

Verifying multiplicative inverses by matrix multiplication

Testing if two matrices with swapped entries are multiplicative inverses

Verifying matrices E and F are multiplicative inverses

In this lesson, we will learn about identity matrices. <a href='[[glossary.html|{"keyword":"Identity matrix"}]]'>Identity matrix</a> is an n by n matrix which all entries <a href='[[glossary.html|{"keyword":"Main Diagonal of a Matrix"}]]'>diagonal</a> from the top left to the bottom right are 1's, and the rest of the entries are 0. There are many types of identity matrices, as listed in the notes section. We will learn how to apply matrix operations with these such as adding, subtracting, and multiplying. Lastly, we will see that identities have a special property. If two matrices are multiplicative inverses, then multiplying them would give an identity matrix.

Identity Matrix

What is an identity matrix?

Scalar multiplication