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

Step 1: Conceptual Foundation:

A matrix \( M \) is defined as symmetric if its transpose \( M^T \) equals \( M \). A matrix \( M \) is skew-symmetric if \( M^T = -M \). Given that matrices \( A \) and \( B \) are skew-symmetric, it follows that \( A^T = -A \) and \( B^T = -B \). The objective is to evaluate the symmetry properties of combinations of these matrices. A pertinent property is that for any skew-symmetric matrix \( M \), \( M^k \) is symmetric when \( k \) is an even integer, and skew-symmetric when \( k \) is an odd integer.

Step 2: Methodology:

The approach will involve applying the rules of matrix transposition and leveraging the characteristic that the parity of the exponent \( k \) determines the symmetry of a skew-symmetric matrix \( M^k \).

Step 3: Detailed Analysis:

1. Assessment of \( A^3 + B^5 \):

Let \( P = A^3 + B^5 \). Its transpose is calculated as follows:

\[ P^T = (A^3 + B^5)^T = (A^3)^T + (B^5)^T = (A^T)^3 + (B^T)^5 \] Substituting \( A^T = -A \) and \( B^T = -B \): \[ P^T = (-A)^3 + (-B)^5 = -A^3 - B^5 = -(A^3 + B^5) = -P \] As \( P^T = -P \), the matrix \( P \) is skew-symmetric. This statement is accurate.

2. Assessment of \( A^{19} \):

Let \( Q = A^{19} \). Since the exponent 19 is odd:

\[ Q^T = (A^{19})^T = (A^T)^{19} = (-A)^{19} = -A^{19} = -Q \] The matrix \( Q \) is skew-symmetric. This statement is accurate.

3. Assessment of \( B^{14} \):

Let \( R = B^{14} \). Since the exponent 14 is even:

\[ R^T = (B^{14})^T = (B^T)^{14} = (-B)^{14} = B^{14} = R \] The matrix \( R \) is symmetric. This statement is accurate.

4. Assessment of \( A^4 + B^5 \):

Let \( S = A^4 + B^5 \). Its transpose is calculated as follows:

\[ S^T = (A^4 + B^5)^T = (A^4)^T + (B^5)^T = (A^T)^4 + (B^T)^5 \] \[ S^T = (-A)^4 + (-B)^5 = A^4 - B^5 \] For \( S \) to be symmetric, \( S^T \) must equal \( S \): \[ A^4 - B^5 = A^4 + B^5 \] This equality simplifies to \( -B^5 = B^5 \), which means \( 2B^5 = 0 \), implying \( B^5 \) must be the zero matrix. This condition is not universally true for all skew-symmetric matrices \( B \). Consequently, \( S \) is not generally symmetric. This statement is inaccurate.

Step 4: Conclusion:

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