Problem 59 - Differential Equations and Linear Algebra

Question

If \(A\) is an \(n \times n\) skew-symmetric matrix and \(n\) is odd, prove that \(\operatorname{det}(A)=0\)

Step-by-Step Solution

Answer

For an odd n, the determinant of a skew-symmetric matrix A is zero because of its properties: \(A^T = -A\) and \(\operatorname{det}(A^T) = \operatorname{det}(A)\). This leads to the equation \((-1)^n\operatorname{det}(A) = \operatorname{det}(A)\), and since n is odd, \((-1)^n=-1\). Thus, \(-1\operatorname{det}(A) = \operatorname{det}(A)\) which implies \(\operatorname{det}(A) = 0\).

Step 1: Note properties of skew-symmetric matrices

Remember that a skew-symmetric matrix has the property that \(A^T = -A\). So, if A is a skew-symmetric matrix, we know that its transpose is the negative of the matrix itself.

Step 2: Utilize transpose and determinant properties

Now, we can use the properties of transpose and determinants. Specifically, we know that \(\operatorname{det}(A^T) = \operatorname{det}(A)\) and \(\operatorname{det}(-A) = (-1)^n\operatorname{det}(A)\).

Step 3: Relating properties of skew-symmetric matrices

Using the properties from step 1 and step 2, \(\operatorname{det}(-A) = \operatorname{det}(A^T) = \operatorname{det}(A)\). Substituting the property of \(\operatorname{det}(-A) = (-1)^n\operatorname{det}(A)\), \((-1)^n\operatorname{det}(A) = \operatorname{det}(A)\).

Step 4: Consider odd n

Since n is odd, we can rewrite \((-1)^n\) as \(-1\). So, \(-1\operatorname{det}(A) = \operatorname{det}(A)\).

Step 5: Solve for det(A)

Now, we can solve for \(\operatorname{det}(A)\) by adding it to both sides of the equation: \((-1+1)\operatorname{det}(A) = 0\).

Step 6: Conclude

From step 5, we find that \(\operatorname{det}(A) = 0\). So, if A is a skew-symmetric matrix with odd dimensions, its determinant must be zero.

Key Concepts

Determinant

The concept of a determinant is essential in understanding matrices. A determinant is a scalar value derived from a square matrix, providing insightful information about the matrix properties, such as solvability and invertibility. For example, a matrix with a determinant of zero is singular, meaning it's not invertible, which implies there is no unique solution to the system of equations it may represent. Determinants can be calculated using various methods, including expansion by minors or special determinant rules for specific types of matrices.
In the context of skew-symmetric matrices, the determinant has unique properties. A skew-symmetric matrix is one where the transpose equals the negative of the original matrix, i.e., if \( A \) is skew-symmetric, then \( A^T = -A \). Crucially, when the order of a skew-symmetric matrix \( n \) is odd, the determinant must be zero. This characteristic arises from the interplay of the skew-symmetric property with the determinant calculation, as explored in our exercise solution.

Matrix Transpose

The transpose of a matrix is a fundamental operation in linear algebra, playing a pivotal role in understanding symmetry, optimization problems, and more. A matrix transpose, denoted as \( A^T \), is achieved by flipping a matrix over its diagonal. This means that row \( i \) becomes column \( i \), and element \( a_{ij} \) in the original matrix becomes element \( a_{ji} \) in the transposed matrix.
Transposes maintain specific properties, such as \( (A^T)^T = A \) and \( (AB)^T = B^T A^T \), which uphold the communicative nature of transposition in matrix operations. Importantly, in a skew-symmetric matrix, the transpose results in the negative of the original matrix \( A^T = -A \). This property is crucial in proving why the determinant of an odd-ordered skew-symmetric matrix is zero as it reflects how transforms don't alter determinant values but do change them based on the matrix's inherent properties.

Properties of Matrices

Matrices have several important properties that make them versatile and powerful in representing linear equations and transformations. Here are some key properties frequently encountered in linear algebra:

• Addition: The sum of two matrices is the matrix obtained by adding corresponding elements. Matrices must be the same size to be added.
• Multiplication: Matrices can be multiplied when the number of columns in the first matrix equals the number of rows in the second. The result is a new matrix.
• Invertibility: A matrix is invertible if there exists another matrix such that their product is the identity matrix. Non-invertible matrices have a determinant of zero, indicating linear dependence.
• Determinant: This scalar value provides insight into matrix properties, such as invertibility and volume scaling in transformations.

Understanding these properties is especially beneficial when dealing with specialized matrices like skew-symmetric ones. These matrices have additional characteristics, such as \( A^T = -A \), leading to unique outcomes — like a zero determinant for odd-ordered matrices, due to the interplay between transposition and matrix order.