Question:
If A and B are symmetric matrices of same order such that \(\text{AB} + \text{BA} = \text{X}\) and \(\text{AB} - \text{BA} = \text{Y}\), then \((\text{XY})^\text{T} =\)
If A and B are symmetric matrices of same order such that \(\text{AB} + \text{BA} = \text{X}\) and \(\text{AB} - \text{BA} = \text{Y}\), then \((\text{XY})^\text{T} =\)
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For any two symmetric matrices \(A\) and \(B\) of the same order, the expression \((AB + BA)\) is always symmetric, while the expression \((AB - BA)\) is always skew-symmetric. Remembering this property cuts your problem-solving time in half!
Updated On: May 25, 2026
- \( \text{XY} \)
- \( \text{X}^\text{T}\text{Y}^\text{T} \)
- \( -\text{YX} \)
- \( -\text{Y}^\text{T}\text{X}^\text{T} \)
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The Correct Option is C
Solution and Explanation
Concept:
The transpose of a matrix satisfies several key operational properties:
• Reversal law for multiplication: \( (M_1 M_2)^\text{T} = M_2^\text{T} M_1^\text{T} \)
• Distributive law over addition/subtraction: \( (M_1 \pm M_2)^\text{T} = M_1^\text{T} \pm M_2^\text{T} \)
• For a symmetric matrix \(M\), \( M^\text{T} = M \)
• For a skew-symmetric matrix \(M\), \( M^\text{T} = -M \) Step 1: Determining the nature of matrices X and Y using given conditions.
Since \(A\) and \(B\) are symmetric matrices, we know that: \[ A^\text{T} = A \quad \text{and} \quad B^\text{T} = B \] Now let's find the transpose of matrix \(X = AB + BA\): \[ X^\text{T} = (AB + BA)^\text{T} = (AB)^\text{T} + (BA)^\text{T} \] Applying the reversal law of transpose: \[ X^\text{T} = B^\text{T}A^\text{T} + A^\text{T}B^\text{T} \] Substituting \(A^\text{T} = A\) and \(B^\text{T} = B\): \[ X^\text{T} = BA + AB = AB + BA = X \] Thus, \(X\) is a symmetric matrix (\(X^\text{T} = X\)). Next, let's find the transpose of matrix \(Y = AB - BA\): \[ Y^\text{T} = (AB - BA)^\text{T} = (AB)^\text{T} - (BA)^\text{T} \] Applying the reversal law of transpose: \[ Y^\text{T} = B^\text{T}A^\text{T} - A^\text{T}B^\text{T} \] Substituting \(A^\text{T} = A\) and \(B^\text{T} = B\): \[ Y^\text{T} = BA - AB = -(AB - BA) = -Y \] Thus, \(Y\) is a skew-symmetric matrix (\(Y^\text{T} = -Y\)).
Step 2: Evaluating the expression \((XY)^\text{T}\).
Using the reversal law of transpose on the product expression \((XY)^\text{T}\): \[ (XY)^\text{T} = Y^\text{T} X^\text{T} \] Substitute the results derived in Step 1 (\(X^\text{T} = X\) and \(Y^\text{T} = -Y\)): \[ (XY)^\text{T} = (-Y)(X) = -YX \]
• Reversal law for multiplication: \( (M_1 M_2)^\text{T} = M_2^\text{T} M_1^\text{T} \)
• Distributive law over addition/subtraction: \( (M_1 \pm M_2)^\text{T} = M_1^\text{T} \pm M_2^\text{T} \)
• For a symmetric matrix \(M\), \( M^\text{T} = M \)
• For a skew-symmetric matrix \(M\), \( M^\text{T} = -M \) Step 1: Determining the nature of matrices X and Y using given conditions.
Since \(A\) and \(B\) are symmetric matrices, we know that: \[ A^\text{T} = A \quad \text{and} \quad B^\text{T} = B \] Now let's find the transpose of matrix \(X = AB + BA\): \[ X^\text{T} = (AB + BA)^\text{T} = (AB)^\text{T} + (BA)^\text{T} \] Applying the reversal law of transpose: \[ X^\text{T} = B^\text{T}A^\text{T} + A^\text{T}B^\text{T} \] Substituting \(A^\text{T} = A\) and \(B^\text{T} = B\): \[ X^\text{T} = BA + AB = AB + BA = X \] Thus, \(X\) is a symmetric matrix (\(X^\text{T} = X\)). Next, let's find the transpose of matrix \(Y = AB - BA\): \[ Y^\text{T} = (AB - BA)^\text{T} = (AB)^\text{T} - (BA)^\text{T} \] Applying the reversal law of transpose: \[ Y^\text{T} = B^\text{T}A^\text{T} - A^\text{T}B^\text{T} \] Substituting \(A^\text{T} = A\) and \(B^\text{T} = B\): \[ Y^\text{T} = BA - AB = -(AB - BA) = -Y \] Thus, \(Y\) is a skew-symmetric matrix (\(Y^\text{T} = -Y\)).
Step 2: Evaluating the expression \((XY)^\text{T}\).
Using the reversal law of transpose on the product expression \((XY)^\text{T}\): \[ (XY)^\text{T} = Y^\text{T} X^\text{T} \] Substitute the results derived in Step 1 (\(X^\text{T} = X\) and \(Y^\text{T} = -Y\)): \[ (XY)^\text{T} = (-Y)(X) = -YX \]
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