Question

The inverse of a symmetric matrix (if it exists) is

• A. a symmetric matrix
• B. a skew-symmetric matrix
• C. a diagonal matrix
• D. none of these

Answer 1

Answer: (A)

Let $A$ be an invertible symmetric matrix.
$\text{ We have }A{A}^{-1}=A^{-1}A=I_n$
$\Rightarrow {\left(A{A}^{-1}\right)}^{\prime }={\left(A^{-1}A\right)}^{\prime }={\left(I_n\right)}^{\prime }$
$\Rightarrow {\left(A^{-1}\right)}^{\prime }{A}^{\prime }=A^{\prime }{\left(A^{-1}\right)}^{\prime }=I_n$
$\Rightarrow {\left(A^{-1}\right)}^{\prime }A=A{\left(A^{-1}\right)}^{\prime }=I_n$
${\left(A^{-1}\right)}^{\prime }=A^{-1}$ [inverse of a matrix is unique]