Which of the following are correct?
(A) If A and B are symmetric matrices such that AB = BA, then AB is symmetric.
(B) If A and B are symmetric matrices of the same order, then (A+B) is a symmetric matrix.
(C) If A and B are symmetric matrices of the same order, then (AB-BA) is a symmetric matrix.
(D) If A and B are symmetric matrices of the same order, then (AB+BA) is a skew symmetric matrix.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (B) only
Since A and B are symmetric, \(A^T = A\) and \(B^T = B\). The reversal law \((XY)^T = Y^TX^T\) gives each transpose.
(A) If A and B are symmetric matrices such that AB = BA, then AB is symmetric. — Correct. Using AB = BA:
\((AB)^T = B^TA^T = BA = AB\)
(B) If A and B are symmetric matrices of the same order, then (A+B) is a symmetric matrix. — Correct. Taking the transpose:
\((A + B)^T = A^T + B^T = A + B\)
(C) If A and B are symmetric matrices of the same order, then (AB-BA) is a symmetric matrix. — Incorrect. It is skew-symmetric:
\((AB - BA)^T = B^TA^T - A^TB^T = BA - AB = -(AB - BA)\)
So AB − BA is skew-symmetric, not symmetric.
(D) If A and B are symmetric matrices of the same order, then (AB+BA) is a skew symmetric matrix. — Incorrect. It is symmetric:
\((AB + BA)^T = BA + AB = AB + BA\)
So AB + BA is symmetric, not skew-symmetric.