If A and B are non-singular symmetric matrices such that `AB=BA`, then prove that `A^(-1) B^(-1)` is symmetric matrix.

Explanation

To prove that \( A^{-1} B^{-1} \) is a symmetric matrix given that \( A \) and \( B \) are non-singular symmetric matrices and \( AB = BA \), we will follow these steps:

### Step 1: Understand the properties of symmetric matrices
Since \( A \) and \( B \) are symmetric matrices, we have:
\[ A^T = A \quad \text{and} \quad B^T = B \]

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