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Master 3x3 Matrix Inverse Using Row Operations | Linear Algebra

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

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Ex.5

In this section, you will learn how to find the inverse of a 3 x 3 matrix. This method requires the use of matrix row operation. The idea is to draw a vertical line in the middle, write the matrix on the left side of the line, and write the 3 x 3 <a href="[[glossary.html|{"keyword":"Identity matrix"}]]">identity matrix</a> on the right side of the line. Then the goal is to do a bunch of matrix row operation so that the identity matrix appears on the left side of the line. Whatever appears on the right side will be the inverse. Note that this method works for 2 x 2 matrices as well, except the right side of the line is a 2 x 2 identity matrix.

The inverse of 3 x 3 matrices with matrix row operations