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Master Cramer's Rule: Solve Linear Systems Efficiently

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

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Intro

Ex.1

Ex.2

Intro to Cramer's Rule for solving a 2x2 linear system

Applying Cramer's Rule to a 3x3 linear system

Solving a 2-variable linear system using Cramer's Rule

Solving a 2x2 system with non-integer solutions using Cramer's Rule

Rearranging equations into standard form before applying Cramer's Rule

Cramer's Rule with an unsolvable 2x2 system (D equals zero)

Solving a 3x3 linear system with Cramer's Rule

Solving a 3-variable system using Cramer's Rule with 3x3 determinants

Rearranging a 3x3 system and applying Cramer's Rule to solve for x, y, z

Last chapter we saw that we are able to solve linear systems with <a href='[[glossary.html|{"keyword":"Gaussian Elimination"}]]'>Gaussian Elimination</a>. Now we are going to take a look at a new method which involves solving linear systems with Cramer's Rule. Cramer's Rule requires us to find the determinant of 2 x 2 and 3 x 3 matrices (depends on your linear system).  However, this rule can only be used if you have the same number of equations and variables. If you have a different number of equations and variables, then finding the determinant will be impossible. Hence, it will not be possible to use Cramer's rule.