Question

If A and B are skew-symmetric matrices, then which one of the following is NOT true?

• A. A3 + B5 is skew-symmetric
• B. A19 is skew-symmetric
• C. B14 is symmetric
• D. A4 + B5 is symmetric

Hint:

Properties of Skew-Symmetric Matrices:

For a skew-symmetric matrix \( M \), the following rules apply:

• \( M^{\text{odd power}} \) is skew-symmetric.
• \( M^{\text{even power}} \) is symmetric.

This rule can help you quickly evaluate the options in such questions.

Answer 1

The Correct Option is D

Concept Definition:

A matrix \( M \) is symmetric if its transpose \( M^T \) equals \( M \). A matrix \( M \) is skew-symmetric if \( M^T = -M \). Given that \( A \) and \( B \) are skew-symmetric (\( A^T = -A \) and \( B^T = -B \)), we will analyze the symmetry of their combinations. A key property for skew-symmetric matrices \( M \) is that \( M^k \) is symmetric for even \( k \) and skew-symmetric for odd \( k \).

Methodology:

We will apply properties of matrix transposition and the established relationship between the exponent \( k \) and the symmetry of a skew-symmetric matrix.

Analysis:

1. \( A^3 + B^5 \) is skew-symmetric:

Let \( P = A^3 + B^5 \). Its transpose is \( P^T = (A^3 + B^5)^T = (A^3)^T + (B^5)^T = (A^T)^3 + (B^T)^5 \). Substituting \( A^T = -A \) and \( B^T = -B \), we get \( P^T = (-A)^3 + (-B)^5 = -A^3 - B^5 = -(A^3 + B^5) = -P \). Since \( P^T = -P \), \( P \) is skew-symmetric. This statement is true.

2. \( A^{19} \) is skew-symmetric:

Let \( Q = A^{19} \). Since the exponent 19 is odd, \( Q^T = (A^{19})^T = (A^T)^{19} = (-A)^{19} = -A^{19} = -Q \). Thus, \( Q \) is skew-symmetric. This statement is true.

3. \( B^{14} \) is symmetric:

Let \( R = B^{14} \). Since the exponent 14 is even, \( R^T = (B^{14})^T = (B^T)^{14} = (-B)^{14} = B^{14} = R \). Thus, \( R \) is symmetric. This statement is true.

4. \( A^4 + B^5 \) is symmetric:

Let \( S = A^4 + B^5 \). Its transpose is \( S^T = (A^4 + B^5)^T = (A^4)^T + (B^5)^T = (A^T)^4 + (B^T)^5 \). Substituting \( A^T = -A \) and \( B^T = -B \), we get \( S^T = (-A)^4 + (-B)^5 = A^4 - B^5 \). For \( S \) to be symmetric, \( S^T \) must equal \( S \), so \( A^4 - B^5 = A^4 + B^5 \). This simplifies to \( -B^5 = B^5 \), or \( 2B^5 = 0 \), implying \( B^5 \) must be the zero matrix. This condition is not universally true for all skew-symmetric matrices \( B \). Therefore, \( S \) is not generally symmetric. This statement is NOT true.

Conclusion:

The statement that is not true is \( A^4 + B^5 \) is symmetric.