Question

The inverse of a symmetric matrix is a:

• A. Diagonal matrix
• B. Scalar matrix
• C. Symmetric matrix
• D. Skew-symmetric matrix

Hint:

Symmetry is preserved under inversion for invertible symmetric matrices. Remember the identity \((A^-1)^T = (A^T)^-1\).

Answer 1

The Correct Option is C

If \(A\) is a symmetric matrix, then \(A^T = A\).
If \(A\) is invertible, we consider \((A^-1)^T\):
\((A^-1)^T = (A^T)^-1\)
Since \(A^T = A\), we have \((A^-1)^T = A^-1\).
This means \(A^-1\) is also symmetric.
Option (A), Diagonal matrix, is symmetric but not all symmetric matrices are diagonal.
Option (B), Scalar matrix, is a special diagonal matrix but is too restrictive.
Option (D), Skew-symmetric matrix, has \(A^T = -A\) and cannot be the inverse of a symmetric matrix in general unless it’s the zero matrix, which is not invertible.
Therefore, the correct answer is (C) Symmetric matrix.