Question

If A and B are symmetric matrices of the same order, which one of the following is not correct?

• A. A + B is a symmetric matrix.
• B. AB + BA is a symmetric matrix.
• C. A + AT and B + BT are symmetric matrices.
• D. AB - BA is a symmetric matrix.

Hint:

The difference of two matrices AB - BA is not symmetric, even if A and B are symmetric matrices.

Answer 1

The Correct Option is D

Topic: Properties of Symmetric Matrices
Understanding the Question: Identify the incorrect statement regarding operations on symmetric matrices $A$ and $B$ ($A^T = A$ and $B^T = B$).
Key Formulas and Approach: Recall $(X+Y)^T = X^T + Y^T$ and $(XY)^T = Y^T X^T$.
Detailed Solution:
Step 1: Check Statement (A). $(A+B)^T = A^T + B^T = A + B$. It is symmetric. Correct.
Step 2: Check Statement (B). $(AB + BA)^T = (AB)^T + (BA)^T = B^T A^T + A^T B^T = BA + AB$. It is symmetric. Correct.
Step 3: Check Statement (C). $(A + A^T)^T = A^T + (A^T)^T = A^T + A$. It is always symmetric. Correct.
Step 4: Check Statement (D). $(AB - BA)^T = (AB)^T - (BA)^T = B^T A^T - A^T B^T = BA - AB$.
Compare this to the original $(AB - BA)$. Since $BA - AB = -(AB - BA)$, this matrix is actually skew-symmetric, not symmetric.
Conclusion: Statement (D) is incorrect.