Question

The inverse of a symmetric matrix is

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

Answer 1

Answer: (B)

Let $A=\left[\begin{array}{ccc}1& -2& 1\\ 2& 3& -5\\ -1& 5& 1\end{array}\right]$ be asked symmetric matrix
$\therefore |A|=1(3+25)+2(2-5)+1(10+3)$
$=28-6+13=35$
$\therefore \mathrm{adj}(A)=\left[\begin{array}{ccc}28& 7& 7\\ 3& 2& 7\\ <br/>13& -3& 7\end{array}\right]$
$\therefore A^{-1}=\frac{1}{35}\left[\begin{array}{ccc}28& 7& 7\\ 3& 2& 7\\ 13& -3& 7\end{array}\right]$
Hence, inverse of a skew symmetric matrix is not skew symmetric matrix.