Answer :
Answer:
Given that \( A \) and \( B \) are symmetric matrices, we need to determine which among the given options are symmetric and which are skew-symmetric:
1. **Symmetry of \( A + B \)**:
\[
(A + B)^T = A^T + B^T = A + B
\]
Therefore, \( A + B \) is symmetric.
2. **Skew-symmetry of \( A - B \)**:
\[
(A - B)^T = A^T - B^T = A - B
\]
Therefore, \( A - B \) is skew-symmetric.
3. **Symmetry of \( AB + BA \)** (assuming multiplication is defined):
\[
(AB + BA)^T = (AB)^T + (BA)^T = B^T A^T + A^T B^T = BA + AB
\]
Since \( AB + BA = AB + BA \), \( AB + BA \) is symmetric.
4. **Skew-symmetry of \( AB - BA \)** (assuming multiplication is defined):
\[
(AB - BA)^T = (AB)^T - (BA)^T = B^T A^T - A^T B^T = - (BA - AB)
\]
Therefore, \( AB - BA \) is skew-symmetric.
**Summary of results**:
- \( A + B \) is **symmetric**.
- \( A - B \) is **skew-symmetric**.
- \( AB + BA \) is **symmetric**.
- \( AB - BA \) is **skew-symmetric**.
So, the correct answers are:
a) \( A + B \) (symmetric)
b) \( A - B \) (skew-symmetric)
c) \( AB + BA \) (symmetric)
d) \( AB - BA \) (skew-symmetric)
Answer:For two symmetric matrices A and B, we can determine the symmetry of the given expressions as follows:
a) **A + B**: The sum of two symmetric matrices is always symmetric. So, A + B is symmetric.
b) **A - B**: The difference of two symmetric matrices is also symmetric. So, A - B is symmetric.
c) **AB + BA**: For any two matrices A and B, (AB)' = B'A'. Since A and B are symmetric, A' = A and B' = B. Therefore, (AB)' = BA, and (BA)' = AB. So, AB + BA is the sum of a matrix and its transpose, which makes it symmetric.
d) **AB - BA**: This expression is known as the commutator of A and B. For symmetric matrices, AB - BA is skew-symmetric. This is because (AB - BA)' = B'A' - A'B' = BA - AB = -(AB - BA).
In summary:
- a) A + B is symmetric.
- b) A - B is symmetric.
- c) AB + BA is symmetric.
- d) AB - BA is skew-symmetric.
Step-by-step explanation: