Problem 70 - Algebra for JEE ADVANCED

Question

If \(A\) is a skew-symmetric matrix and \(n\) is odd positive integer, then \(A^{n}\) is a. a skew-symmetric matrix b. a symmetric matrix c. a diagonal matrix d. none of these

Step-by-Step Solution

Answer

The answer is a. a skew-symmetric matrix.

Step 1: Define Skew-Symmetric Matrix

A matrix \( A \) is called skew-symmetric if \( A^T = -A \), where \( A^T \) is the transpose of \( A \). This means the elements satisfy \( a_{ij} = -a_{ji} \) for all \( i \) and \( j \), and all the diagonal elements are zero.

Step 2: Property of Skew-Symmetric Matrices Raised to Odd Power

Skew-symmetric matrices raised to an odd power remain skew-symmetric. Specifically, since \( A \) is skew-symmetric, for any odd positive integer \( n \), we have \( (A^n)^T = (A^T)^n = (-A)^n = -A^n \). This confirms that \( A^n \) is still skew-symmetric.

Step 3: Conclusion on Matrix Type

As demonstrated in Step 2, since \( A^n \) satisfies \( (A^n)^T = -A^n \), it is skew-symmetric by definition. Therefore, \( A^n \) is a skew-symmetric matrix when \( n \) is an odd positive integer.

Key Concepts

Understanding Matrix Transpose

A transpose of a matrix is obtained by flipping a matrix over its diagonal. This means rows become columns and vice versa. Given a matrix \( A \), the transpose, denoted as \( A^T \), has elements such that the element in the \(i\)-th row and \(j\)-th column of \( A \) becomes the \(j\)-th row and \(i\)-th column in \( A^T \).

• For example, if \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), then \( A^T = \begin{bmatrix} a & c \ b & d \end{bmatrix} \).
• This property is crucial in identifying skew-symmetric matrices, where \( A^T = -A \).

In skew-symmetric matrices, this means each off-diagonal element changes sign. Additionally, the diagonal elements must be zero because equating them to their negatives gives zero. Understanding the transpose is vital for exploring the properties of skew-symmetric matrices.

Exploring Odd Powers

Odd powers refer to raising a number or a matrix to an odd exponent. For a skew-symmetric matrix \( A \), if you raise it to an odd positive integer \( n \), the resulting matrix \( A^n \) retains its skew-symmetric property. This is because:

• When you take the transpose of \( A^n \), which is \( (A^n)^T \), it equals \( (A^T)^n \).
• Since \( A^T = -A \), then \( (A^T)^n = (-A)^n = -A^n \).

Raising a skew-symmetric matrix to any odd power gives a matrix where the property \( (A^n)^T = -A^n \) holds true. This is because an odd number of negative factors results in a negative product, preserving the skew-symmetry. This concept is important as it helps in concluding the nature of certain matrix operations.

Algebraic Properties of Skew-Symmetric Matrices

Skew-symmetric matrices have unique algebraic properties that make them intriguing in linear algebra. A skew-symmetric matrix is defined such that \( A^T = -A \). Let's explore some properties:

• Any skew-symmetric matrix must have zeros on its diagonal. This occurs because the diagonal elements \( a_{ii} = -a_{ii} \), which means \( a_{ii} = 0 \).
• When raised to an odd power, as established, they remain skew-symmetric, which is not the case for even powers.
• Two skew-symmetric matrices can be added to produce another skew-symmetric matrix.

These properties allow skew-symmetric matrices to be used in various mathematical and applied fields, including physics and computer science. They offer insights into the behavior of matrix operations, especially in terms of symmetry and inversion, showcasing a relationship between the structure of the matrix and its algebraic operations.