This document contains numerous examples and solutions related to matrices, determinants, vectors, and linear algebra concepts such as orthogonality, independence, diagonalization, and inverse matrices. Each example provides the setup, working, and solution for mathematical expressions and equations involving matrices, vectors, and determinants. No overarching topic or theme is discussed across the examples.

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Example
Laplace’s Equation
The Laplace transform

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Example
Solution
From
We have

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Example
Find the inverse Laplace transform of
Solution
Expanding the Laplace transform in terms of partial fractions gives

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Using the result
We have
Example

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Solution
Example
Solution
By definition

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Mathematical Methods in Physics
204 Matrices, orthogonal and unitary matrices, matrix diagonalization
205 Neumann Functions
206 Orthogonal curvilinear coordinates system
207 Orthogonal functions
208 Orthogonality
Matrix & Determinants
Example
Solution
Characteristic equation is

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Example
Solution

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Example
Example
Multiplication of a Matrix and a Vector
Example

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Products of Row and Column Vectors
Example
Example
Transposition of Matrices and Vectors
If
A little more compactly, we can write

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And
Rules for transposition are
Example
Symmetric and Skew-Symmetric Matrices

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Example
Upper and Lower Triangular Matrices
Example
Diagonal Matrix D. Scalar Matrix S. Unit Matrix I
Example
Linear Independence and Dependence
The three vectors

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are linearly dependent because
Example
Cramer’s Rule for Two Equations
Example
Expansions of a Third-Order Determinant

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Example
Determinant of a Triangular Matrix
Example

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Example
The matrices

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are symmetric, skew-symmetric, and orthogonal, respectively.
Example
Notations
Hermitian, Skew-Hermitian, and Unitary Matrices
Example
Hermitian, Skew-Hermitian, and Unitary Matrices

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are Hermitian, skew-Hermitian, and unitary matrices, respectively.
Example

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Example
Example
But

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Example
Example
Example

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Orthogonal Matrices

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Hermitian and Skew-Hermitian Matrices
Conjugation (the operation of taking a conjugate) and magnitude have the following
properties.

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Norm of a Vector
Orthogonal Vectors
Example
Let

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Then
Example
Let
Then
And

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Example
But
Example
Consistent System of Equations
Matrix Inverse

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Nonsingular and Singular Matrices
Example
Let

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Just for illustration, expand by row 3:
Consider again
If we expand by column 2 we get

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Example