If A and B are symmetric matrices of the same order then (A) A-B is skew symmetric (B) A+B is symmetric (C) AB-BA is skew symmetric (D) AB+BA is symmetric

Options

A. skew symmetric matrix
B. symmetric symmetric matrix
C. zero matrix
D. identity matrix

Explanation

To solve the problem, we need to analyze the properties of symmetric and skew-symmetric matrices. Given that \( A \) and \( B \) are symmetric matrices, we know that:

\[ A^T = A \quad \text{and} \quad B^T = B \]

We will evaluate each option one by one.

### Option A: \( A - B \) is skew-symmetric

To check if \( A - B \) is skew-symmetric, we need to find the transpose of \( A - B \):

\[ (A - B)^T = A^T - B^T \]

Using the property of symmetric matrices:

\[ = A - B \]

Since \( (A - B)^T = A - B \), this means \( A - B \) is symmetric, not skew-symmetric. Therefore, **Option A is incorrect**.

### Option B: \( A + B \) is symmetric

Now, we check if \( A + B \) is symmetric:

\[ (A + B)^T = A^T + B^T \]

Again, using the property of symmetric matrices:

\[ = A + B \]

Since \( (A + B)^T = A + B \), this means \( A + B \) is symmetric. Therefore, **Option B is correct**.

### Option C: \( AB - BA \) is skew-symmetric

Next, we check if \( AB - BA \) is skew-symmetric:

\[ (AB - BA)^T = (AB)^T - (BA)^T \]

Using the property of transpose of a product:

\[ = B^T A^T - A^T B^T \]

Substituting the values of \( A^T \) and \( B^T \):

\[ = BA - AB \]

This can be rewritten as:

\[ = -(AB - BA) \]

Since \( (AB - BA)^T = -(AB - BA) \), this means \( AB - BA \) is skew-symmetric. Therefore, **Option C is correct**.

### Option D: \( AB + BA \) is symmetric

Finally, we check if \( AB + BA \) is symmetric:

\[ (AB + BA)^T = (AB)^T + (BA)^T \]

Using the property of transpose of a product:

\[ = B^T A^T + A^T B^T \]

Substituting the values of \( A^T \) and \( B^T \):

\[ = BA + AB \]

Since \( (AB + BA)^T = AB + BA \), this means \( AB + BA \) is symmetric. Therefore, **Option D is correct**.

### Conclusion

The correct options are:
- **Option B**: \( A + B \) is symmetric.
- **Option C**: \( AB - BA \) is skew-symmetric.
- **Option D**: \( AB + BA \) is symmetric.