Problem 39 - Advanced Engineering Mathematics

Question

An \(n \times n\) matrix \(\mathbf{A}\) is said to be skew-symmetric if \(\mathbf{A}^{T}=-\mathbf{A}\). If \(\mathbf{A}\) is a \(5 \times 5\) skew-symmetric matrix, show that \(\operatorname{det} \mathbf{A}=0\).

Step-by-Step Solution

Answer

The determinant of a skew-symmetric matrix of odd order is zero.

Step 1: Understand the Property of Skew-Symmetric Matrices

A matrix \( \mathbf{A} \) is skew-symmetric if the transpose of \( \mathbf{A} \) is equal to its negative: \( \mathbf{A}^{T} = -\mathbf{A} \). In a skew-symmetric matrix, all diagonal elements are zero.

Step 2: Explore Determinant Properties

For any square matrix \( \mathbf{A} \), the determinant of the transpose is equal to the determinant of the original matrix: \( \operatorname{det}(\mathbf{A}^T) = \operatorname{det}(\mathbf{A}) \). Additionally, for any matrix \( \mathbf{A} \), the determinant of \(-\mathbf{A}\) is \((-1)^n \operatorname{det}(\mathbf{A}) \), where \(n\) is the order of the matrix.

Step 3: Apply Determinant Property to Skew-Symmetric Matrix

Given that \( \mathbf{A}^{T} = -\mathbf{A} \), we examine the determinants: \( \operatorname{det}(\mathbf{A}^{T}) = \operatorname{det}(-\mathbf{A}) = (-1)^5 \operatorname{det}(\mathbf{A}) = -\operatorname{det}(\mathbf{A}) \).

Step 4: Conclude Determinant Value

Since we know \( \operatorname{det}(\mathbf{A}^{T}) = \operatorname{det}(\mathbf{A}) \), replace \( \operatorname{det}(\mathbf{A}) \) by \(-\operatorname{det}(\mathbf{A}) \) as shown in the previous step and imply that \( \operatorname{det}(\mathbf{A}) = -\operatorname{det}(\mathbf{A}) \). This equation can only hold true if \( \operatorname{det}(\mathbf{A}) = 0 \).

Key Concepts

Determinant

The determinant is a special number associated with square matrices, which can be denoted as \( \operatorname{det}(\mathbf{A}) \), where \( \mathbf{A} \) is a matrix. It serves as a useful property in linear transformations and provides valuable matrix insights.
In essence, the determinant helps to:

• Determine if a matrix is invertible
• Find the volume distortion factor of linear transformations described by the matrix
• Evaluate the eigenvalues of a matrix

The determinant of a matrix also { negates when the matrix is multiplied by -1. For example, if \( \mathbf{A} \) is a matrix, then \( \operatorname{det}(-\mathbf{A}) = (-1)^n \operatorname{det}(\mathbf{A}) \), where \( n \) refers to the dimensions of the square matrix.
This property plays a crucial role, especially for skew-symmetric matrices, as it can help demonstrate why some determinants equal zero in such contexts.

Matrix Transpose

The transpose of a matrix, denoted as \( \mathbf{A}^T \), is achieved by flipping the matrix over its diagonal. This means that the rows become columns and vice versa.
For example, if your matrix \( \mathbf{A} \) is:
\[\begin{bmatrix}a & b \c & d\end{bmatrix}\]
The transpose \( \mathbf{A}^T \) would be:
\[\begin{bmatrix}a & c \b & d\end{bmatrix}\]
Understanding the transpose is crucial in grasping matrix properties and behaviors with respect to operations like matrix multiplication.
A key fact about transpose operations in skew-symmetric matrices is that \( \mathbf{A}^T = -\mathbf{A} \). This leads to specialized characteristics and applications in both theoretical and practical aspects of linear algebra.

Matrix Properties

Understanding matrix properties is essential in navigating various linear algebra challenges. Key properties reveal significant insights into matrix behavior.
Skew-Symmetric Matrix:

• A matrix where \( \mathbf{A}^T = -\mathbf{A} \) and all diagonal elements are zero
• For odd-dimensional skew-symmetric matrices (like a 5x5 matrix), the determinant is always zero, indicating singularity

Determinant Properties:

• The determinant of a transpose equals the determinant of the original matrix: \( \operatorname{det}(\mathbf{A}^T) = \operatorname{det}(\mathbf{A}) \)
• If multiplying the matrix by a scalar \(-1\), the determinant is then multiplied by \((-1)^n\)

Recognizing these characteristics:

• Provides better insights on linear transformations and effects on coordinate systems
• Clarifies why certain matrices lead to specific outcomes, such as non-invertibility or volume reduction

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, linear transformations, and their representations in vector spaces and matrices. The field is foundational, with widespread applications across science and engineering.
Key components of linear algebra include:

• Matrices and determinants, highlighting matrix operations and solving systems of linear equations
• Vector spaces, focusing on the structures and properties formed by vector operations
• Applications in engineering and physics, where systems often require solving complex equations through linear representations

Linear algebra facilitates a deeper understanding of concepts such as matrix ranks, linear mappings, eigenvectors, and eigenvalues.
By understanding linear algebra's core principles, one can appreciate their practical utility, such as dynamic system modeling and computer graphics transformations. Moreover, recognizing concepts like skew-symmetric matrices and their properties, leads to advanced mathematical and scientific insights.