Question

If A and B are symmetric matrices of same order then
A. AB is symmetric iff $AB=BA$

B. $AB+BA$ is skew symmetric matrix

C. AB-BA is symmetric matrix

D. $(A+B)^{n}$ is symmetric for all $n \in N$

Choose the correct answer from the options given below

• A. A, B only
• B. B, C only
• C. C, D only
• D. A, D only

Hint:

Always remember the property $(AB)^T = B^T A^T$. If two symmetric matrices commute ($AB=BA$), their product remains symmetric. If they do not, their product is generally neither symmetric nor skew-symmetric.

Answer 1

The Correct Option is D

We are given that A and B are symmetric matrices, which means $A^T = A$ and $B^T = B$.
We need to evaluate the properties of their products and sums.

Step 1: \color{redEvaluate Statement A
For AB to be symmetric, $(AB)^T$ must equal $AB$.
Using the reversal law of transposes: $(AB)^T = B^T A^T$.
Since A and B are symmetric, $B^T A^T = BA$.
Therefore, $(AB)^T = AB$ if and only if $BA = AB$.
This confirms that statement A is correct.

Step 2: \color{redEvaluate Statements B and C
For $AB+BA$: $(AB+BA)^T = (AB)^T + (BA)^T = B^T A^T + A^T B^T = BA + AB$.
Since $(AB+BA)^T = AB+BA$, it is a symmetric matrix, not skew-symmetric.
Thus, statement B is incorrect.
For $AB-BA$: $(AB-BA)^T = (AB)^T - (BA)^T = BA - AB = -(AB - BA)$.
Since $(AB-BA)^T = -(AB-BA)$, it is a skew-symmetric matrix, not symmetric.
Thus, statement C is incorrect.

Step 3: \color{redEvaluate Statement D
Let $S = A+B$. Since $A^T=A$ and $B^T=B$, $S^T = (A+B)^T = A^T + B^T = A+B = S$.
So, $(A+B)$ is symmetric.
Now consider $S^n$. We know $(S^n)^T = (S^T)^n$.
Since $S^T = S$, then $(S^T)^n = S^n$.
Therefore, $(A+B)^n$ is symmetric for any natural number $n$.
This confirms statement D is correct.

Step 4: \color{redConclusion
Statements A and D are correct.
This matches Option (4).