The inverse of a symmetric matrix is a matrix which is

Options

A. diagonal
B. symmetric
C. skew symmetric
D. None of these

Answer

To solve the question regarding the inverse of a symmetric matrix, we can follow these steps:

### Step-by-Step Solution:

1. **Definition of Symmetric Matrix**:
A matrix \( A \) is symmetric if \( A^T = A \), where \( A^T \) is the transpose of matrix \( A \).

**Hint**: Remember that the transpose of a matrix is obtained by swapping its rows and columns.

2. **Taking the Inverse**:
We want to find the inverse of a symmetric matrix \( A \). Let’s denote the inverse of \( A \) as \( A^{-1} \).

**Hint**: The inverse of a matrix \( A \) is defined such that \( A A^{-1} = I \), where \( I \) is the identity matrix.

3. **Transpose of the Inverse**:
We can use the property of transposes that states \( (A^{-1})^T = (A^T)^{-1} \). Since \( A \) is symmetric, we have \( A^T = A \). Therefore, we can write:
\[ (A^{-1})^T = (A^T)^{-1} = A^{-1} \]

**Hint**: This property shows that the transpose of the inverse of a matrix is equal to the inverse of the transpose of that matrix.

4. **Conclusion**:
From the equation \( (A^{-1})^T = A^{-1} \), we can conclude that \( A^{-1} \) is symmetric because it satisfies the condition \( (A^{-1})^T = A^{-1} \).

**Hint**: A matrix is symmetric if it is equal to its own transpose.

### Final Answer:
The inverse of a symmetric matrix is also a symmetric matrix.