Question

If A is a non-identity invertible symmetric matrix, then \(A^{-1}\) is:

• A. Symmetric matrix
• B. Skew-symmetric matrix
• C. Identity matrix
• D. Zero matrix

Answer 1

The Correct Option is A

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:

\(A = A^T\)

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

\((A^{-1})^T = (A^T)^{-1}\)

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

\((A^{-1})^T = A^{-1}\)

This implies that A^-1 is also symmetric.

Thus, the correct answer is: Symmetric matrix.