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: Understanding the Concept:

A matrix \( M \) is symmetric if \( M^T = M \).
A matrix \( M \) is skew-symmetric if \( M^T = -M \).
We are given that \( A \) and \( B \) are skew-symmetric, so \( A^T = -A \) and \( B^T = -B \). We need to check the properties of combinations of these matrices. A useful property is that for a skew-symmetric matrix \( M \), \( M^k \) is symmetric if \( k \) is even, and skew-symmetric if \( k \) is odd.

Step 2: Key Formula or Approach:

We will use the properties of matrix transposition and the fact that for skew-symmetric matrices, the power \( k \) determines the symmetry of the matrix.

Step 3: Detailed Explanation:

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

Let \( P = A^3 + B^5 \). Let's find its transpose:

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

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

Let \( Q = A^{19} \). The power 19 is odd:

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

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

Let \( R = B^{14} \). The power 14 is even:

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

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

Let \( S = A^4 + B^5 \). Let's find its transpose:

\[ S^T = (A^4 + B^5)^T = (A^4)^T + (B^5)^T = (A^T)^4 + (B^T)^5 \] \[ S^T = (-A)^4 + (-B)^5 = A^4 - B^5 \] For \( S \) to be symmetric, \( S^T \) must be equal to \( S \): \[ A^4 - B^5 = A^4 + B^5 \] This implies \( -B^5 = B^5 \), or \( 2B^5 = 0 \), which means \( B^5 \) must be the zero matrix. This is not true for all skew-symmetric matrices \( B \). Therefore, \( S \) is not generally symmetric. This statement is NOT true.

Step 4: Final Answer:

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