If A and B are two symmetric matrices of the same order, then
Match List-I with List-II
| List-I | List-II | ||
| (A) | \( (\mathrm{AB})^{\mathrm{T}} \) | (I) | AB |
| (B) | AB − BA | (II) | Symmetric matrix |
| (C) | \( (\mathrm{BA})^{\mathrm{T}} \) | (III) | Skew-symmetric matrix |
| (D) | AB + BA | (IV) | BA |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) - (IV), (B) - (III), (C) - (I), (D) - (II)
By the reversal law \( (\mathrm{XY})^{\mathrm{T}}=\mathrm{Y}^{\mathrm{T}}\mathrm{X}^{\mathrm{T}} \), and since A and B are symmetric, \( \mathrm{A}^{\mathrm{T}}=\mathrm{A} \) and \( \mathrm{B}^{\mathrm{T}}=\mathrm{B} \).
(A) \( (\mathrm{AB})^{\mathrm{T}} \)
This matches (IV) BA.
\( \displaystyle (\mathrm{AB})^{\mathrm{T}}=\mathrm{B}^{\mathrm{T}}\mathrm{A}^{\mathrm{T}}=\mathrm{BA} \)
(B) AB − BA
This matches (III) Skew-symmetric matrix.
\( \displaystyle (\mathrm{AB}-\mathrm{BA})^{\mathrm{T}}=\mathrm{BA}-\mathrm{AB} \)
\( \displaystyle =-(\mathrm{AB}-\mathrm{BA}) \)
(C) \( (\mathrm{BA})^{\mathrm{T}} \)
This matches (I) AB.
\( \displaystyle (\mathrm{BA})^{\mathrm{T}}=\mathrm{A}^{\mathrm{T}}\mathrm{B}^{\mathrm{T}}=\mathrm{AB} \)
(D) AB + BA
This matches (II) Symmetric matrix.
\( \displaystyle (\mathrm{AB}+\mathrm{BA})^{\mathrm{T}}=\mathrm{BA}+\mathrm{AB} \)
\( \displaystyle =\mathrm{AB}+\mathrm{BA} \)
So the matching is (A)-(IV), (B)-(III), (C)-(I), (D)-(II).