Invertible Matrices

A matrix A is symmetric if A = A^T, where A^T is the transpose of A. Given that A is a non-identity, invertible symmetric matrix, we need to determine the nature of A^-1.

To prove that A^-1 is symmetric, we start with the given property of symmetric matrices:

Since A is invertible, A^-1 exists. Taking the inverse of both sides of the equation A = A^T, we have:

Since A is symmetric, A = A^T, so:

This implies that A^-1 is also symmetric.

Thus, the correct answer is: Symmetric matrix.