Question:
If \(A\) and \(B\) are skew-symmetric matrices of same order, then \(AB' + BA'\) is a/an :
If \(A\) and \(B\) are skew-symmetric matrices of same order, then \(AB' + BA'\) is a/an :
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The transpose of a product always reverses the order of the matrices.
Even though \(A\) and \(B\) are individually skew-symmetric, specific combinations like \(AB' + BA'\) result in a symmetric structure because the signs cancel out during the transpose operation.
Updated On: Sep 10, 2026
- symmetric matrix
- skew-symmetric matrix
- null matrix
- identity matrix
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The Correct Option is A
Solution and Explanation
Concept:
• A matrix \(M\) is symmetric if \(M' = M\).
• A matrix \(M\) is skew-symmetric if \(M' = -M\).
• Transpose properties to use: \((A+B)' = A' + B'\), \((AB)' = B'A'\), and \((A')' = A\).
Step 1: Apply the skew-symmetric definitions
Since \(A\) and \(B\) are skew-symmetric matrices: \[ A' = -A \] \[ B' = -B \]
Step 2: Analyze the target matrix by taking its transpose
Let \(X = AB' + BA'\). To find the nature of \(X\), we compute \(X'\): \[ X' = (AB' + BA')' \] Using the addition rule for transposes: \[ X' = (AB')' + (BA')' \]
Step 3: Apply the product rule for transposes
Using the rule \((MN)' = N'M'\): \[ X' = (B')'A' + (A')'B' \] Since the double transpose of a matrix is the matrix itself (\((M')' = M\)): \[ X' = BA' + AB' \]
Step 4: Compare with the original expression
Matrix addition is commutative (\(P + Q = Q + P\)): \[ X' = AB' + BA' \] We observe that \(X' = X\). By definition, this means the matrix is symmetric.
• A matrix \(M\) is symmetric if \(M' = M\).
• A matrix \(M\) is skew-symmetric if \(M' = -M\).
• Transpose properties to use: \((A+B)' = A' + B'\), \((AB)' = B'A'\), and \((A')' = A\).
Step 1: Apply the skew-symmetric definitions
Since \(A\) and \(B\) are skew-symmetric matrices: \[ A' = -A \] \[ B' = -B \]
Step 2: Analyze the target matrix by taking its transpose
Let \(X = AB' + BA'\). To find the nature of \(X\), we compute \(X'\): \[ X' = (AB' + BA')' \] Using the addition rule for transposes: \[ X' = (AB')' + (BA')' \]
Step 3: Apply the product rule for transposes
Using the rule \((MN)' = N'M'\): \[ X' = (B')'A' + (A')'B' \] Since the double transpose of a matrix is the matrix itself (\((M')' = M\)): \[ X' = BA' + AB' \]
Step 4: Compare with the original expression
Matrix addition is commutative (\(P + Q = Q + P\)): \[ X' = AB' + BA' \] We observe that \(X' = X\). By definition, this means the matrix is symmetric.
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