Question:
If A and B are symmetric matrics of same order, then \( (AB - BA) \) is a
If A and B are symmetric matrics of same order, then \( (AB - BA) \) is a
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Tip 1: For any symmetric matrices \( A, B \):
Tip 2: \( AB + BA \) is always symmetric.
Tip 3: \( AB - BA \) is always skew-symmetric.
Tip 2: \( AB + BA \) is always symmetric.
Tip 3: \( AB - BA \) is always skew-symmetric.
Updated On: Sep 10, 2026
- Zero matrix
- Identity matrix
- Symmetric matrix
- Skew symmetric matrix
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The Correct Option is D
Solution and Explanation
Concept:
• Since \( A \) and \( B \) are symmetric, \( A^T = A \) and \( B^T = B \).
• A matrix \( C \) is symmetric if \( C^T = C \).
• A matrix \( C \) is skew-symmetric if \( C^T = -C \).
Step 1: Let the resulting matrix be \( C \)
Define \( C = AB - BA \). To check its property, we must find its transpose \( C^T \).
Step 2: Find the transpose using properties
\[ C^T = (AB - BA)^T \] \[ C^T = (AB)^T - (BA)^T \] Apply the reversal law \( (XY)^T = Y^T X^T \): \[ C^T = B^T A^T - A^T B^T \]
Step 3: Substitute the given conditions
Since \( A^T = A \) and \( B^T = B \): \[ C^T = BA - AB \]
Step 4: Compare \( C^T \) with \( C \)
Factor out a negative sign: \[ C^T = -(AB - BA) \] \[ C^T = -C \] Since \( C^T = -C \), the matrix is skew-symmetric.
• Since \( A \) and \( B \) are symmetric, \( A^T = A \) and \( B^T = B \).
• A matrix \( C \) is symmetric if \( C^T = C \).
• A matrix \( C \) is skew-symmetric if \( C^T = -C \).
Step 1: Let the resulting matrix be \( C \)
Define \( C = AB - BA \). To check its property, we must find its transpose \( C^T \).
Step 2: Find the transpose using properties
\[ C^T = (AB - BA)^T \] \[ C^T = (AB)^T - (BA)^T \] Apply the reversal law \( (XY)^T = Y^T X^T \): \[ C^T = B^T A^T - A^T B^T \]
Step 3: Substitute the given conditions
Since \( A^T = A \) and \( B^T = B \): \[ C^T = BA - AB \]
Step 4: Compare \( C^T \) with \( C \)
Factor out a negative sign: \[ C^T = -(AB - BA) \] \[ C^T = -C \] Since \( C^T = -C \), the matrix is skew-symmetric.
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