Matrices Cheat Sheet - Determinants, Inverses, Eigenvalues, and Applications

Matrices - Matrices are a powerful mathematical tool used to solve systems of linear equations, represent transformations, and model complex systems in various fields such as engineering, physics, computer science, and economics. This cheat sheet provides a comprehensive overview of the essential concepts, properties, and operations related to matrices.

Basics of Matrices

1. Definition

A matrix is a rectangular array of numbers arranged in rows and columns. A matrix of size $m \times n$ has $m$ rows and $n$ columns.

2. Types of Matrices

• Square Matrix: Rows and columns are equal ($m = n$).
• Identity Matrix ($I$): A diagonal matrix with 1s along the diagonal and 0s elsewhere.
• Zero Matrix ($0$): All elements are 0.
• Diagonal Matrix: Non-zero elements only on the diagonal.
• Symmetric Matrix: $A = A^T$, where $A^T$ is the transpose of $A$.

Operations on Matrices

1. Addition and Subtraction

Matrices can be added or subtracted if they have the same dimensions:

• $C = A + B$ or $C = A - B$

2. Scalar Multiplication

Each element of a matrix is multiplied by a scalar:

• $C = kA$, where $k$ is a scalar.

3. Matrix Multiplication

The product of two matrices $A$ ($m \times n$) and $B$ ($n \times p$) results in a matrix $C$ ($m \times p$):

• $C_{ij} = \sum_{k=1}^n A_{ik}B_{kj}$

4. Transpose

The transpose of a matrix $A$ is obtained by flipping it over its diagonal:

• $A^T$: Rows become columns.

Determinants and Inverses

1. Determinant

The determinant of a square matrix measures the matrix's invertibility:

• For a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$: $\det(A) = ad - bc$
• For a $3 \times 3$ matrix, expand using minors and cofactors.

2. Inverse

The inverse of a square matrix $A$ exists if $\det(A) \neq 0$:

• $A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$, where $\text{adj}(A)$ is the adjugate matrix.

3. Rank

The rank of a matrix is the maximum number of linearly independent rows or columns.

Eigenvalues and Eigenvectors

1. Definition

For a square matrix $A$, if $A \mathbf{v} = \lambda \mathbf{v}$, then:

• $\lambda$: Eigenvalue
• $\mathbf{v}$: Eigenvector

2. Finding Eigenvalues

Solve the characteristic equation:

• $\det(A - \lambda I) = 0$

3. Finding Eigenvectors

Substitute each eigenvalue into $(A - \lambda I)\mathbf{v} = 0$ to find $\mathbf{v}$.

Solving Linear Systems Using Matrices

1. Linear System Representation

A system of linear equations can be represented as:

• $A \mathbf{x} = \mathbf{b}$, where $A$ is the coefficient matrix, $\mathbf{x}$ is the variable vector, and $\mathbf{b}$ is the constant vector.

2. Gaussian Elimination

This method involves row operations to reduce $A$ to upper triangular form, simplifying the solution process.

3. Matrix Inversion Method

If $A$ is invertible:

• $\mathbf{x} = A^{-1} \mathbf{b}$

Applications of Matrices

• Transformations: Rotations, translations, and scaling in graphics.
• Solving Linear Systems: Used in engineering and physics for solving equations.
• Data Representation: Representing graphs, networks, and datasets.
• Eigenvalue Problems: Modeling vibrations, stability, and quantum mechanics.

Summary Table of Key Equations

Concept	Equation
Determinant (2x2)	$\det(A) = ad - bc$
Inverse	$A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$
Eigenvalue Equation	$\det(A - \lambda I) = 0$
Matrix Multiplication	$C_{ij} = \sum_{k=1}^n A_{ik}B_{kj}$

FAQs About Matrices

Q1: What is the determinant of a matrix used for?

The determinant helps determine if a matrix is invertible and provides insights into transformations such as scaling and rotations.

Q2: How do you calculate the inverse of a matrix?

For a square matrix, the inverse is calculated using $A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$, provided $\det(A) \neq 0$.

Q3: What are eigenvalues and eigenvectors used for?

They are used in various fields like vibration analysis, stability studies, and principal component analysis in data science.

Q4: How does Gaussian elimination solve linear systems?

Gaussian elimination reduces the coefficient matrix to an upper triangular form, making it easier to solve the equations through back substitution.

Q5: What is the rank of a matrix?

The rank is the number of linearly independent rows or columns in a matrix and indicates the dimension of the vector space spanned by its rows or columns.