Problem 30 - A Complete Resource Book in Mathematics for JEE Main

Question

The inverse of a skew-symmetric matrix of odd order is (A) a symmetric matrix (B) a skew-symmetric matrix (C) diagonal matrix (D) does not exist

Step-by-Step Solution

Answer

The inverse does not exist since the matrix is singular.

Step 1: Define a Skew-Symmetric Matrix

A skew-symmetric matrix is a square matrix \(A\) where \(A^T = -A\). This means that the transpose of the matrix is equal to the negative of the matrix itself.

Step 2: Properties of Skew-Symmetric Matrices

For a skew-symmetric matrix of order \(n\), where \(n\) is odd, all the diagonal elements must be zero. This is due to the property \(a_{ii} = -a_{ii}\), which implies \(a_{ii} = 0\).

Step 3: Determinant of a Skew-Symmetric Matrix

The determinant of a skew-symmetric matrix is zero when the order \(n\) is odd. This is because the determinant is the product of the eigenvalues, and for an odd order, it must include zero as an eigenvalue. Thus, \(|A| = 0\).

Step 4: Consequences of a Zero Determinant

If the determinant of a matrix is zero, then the matrix is not invertible. An inverse of a matrix only exists if its determinant is non-zero.

Step 5: Conclude Based on Determinant

Since the determinant of a skew-symmetric matrix of odd order is zero, the matrix doesn't have an inverse. Thus, the answer to the problem is that the inverse does not exist.

Key Concepts

Matrix Invertibility

Matrix invertibility is a key concept in linear algebra that determines whether a matrix has an inverse. A matrix is invertible, or non-singular, if it has an inverse. This means there exists another matrix that, when multiplied with the original, results in the identity matrix. The identity matrix acts like the number 1 in real numbers; multiplying any matrix by the identity matrix leaves it unchanged.
For a square matrix to be invertible, several conditions must be met:

• The matrix must be square, meaning it has the same number of rows and columns.
• The determinant of the matrix must be non-zero. If the determinant is zero, the matrix is said to be singular and does not have an inverse.

Understanding whether a matrix is invertible involves checking these criteria, particularly focusing on the determinant. For skew-symmetric matrices, especially of odd order, the determinant plays a central role in determining invertibility.

Determinant Properties

The determinant is a scalar value that is a key property of a square matrix. It provides important information about the matrix, particularly in relation to its invertibility and the volume transformation described by the matrix.

• A matrix with a zero determinant is not invertible. This is crucial because it indicates that the matrix cannot be reversed, meaning there's no unique matrix that can transform it back to the identity matrix.
• The determinant can be thought of as the scaling factor for the linear transformation represented by the matrix. A zero value means that the transformation collapses the space into a lower dimension, losing some information about the original space.

For skew-symmetric matrices of odd order, the determinant is always zero. This is due to the presence of a zero eigenvalue, which indicates that the transformation is collapsing dimensions and thus is non-invertible.

Eigenvalues

Eigenvalues are numbers that are fundamental to understanding the behavior of matrices. They play a critical role in a wide range of matrix-related problems. The eigenvalues of a matrix are solutions to the equation \(A - \lambda I = 0\), where \(\lambda\) represents the eigenvalues and \(I\) is the identity matrix.

• Each eigenvalue corresponds to an eigenvector, which describes the directions that are invariant under the linear transformation represented by the matrix.
• For skew-symmetric matrices, eigenvalues have special properties. If the matrix is of odd order, at least one of the eigenvalues must be zero.

This zero eigenvalue results in a determinant of zero, confirming that such a matrix is not invertible. Eigenvalues thus provide insight into the transformation properties of matrices and are crucial in determining invertibility, especially in skew-symmetric matrices.