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Q. The inverse of a symmetric matrix is

AMUAMU 2012Determinants

Solution:

Let $A= \begin{bmatrix}1 & -2 & 1 \\ 2 & 3 & -5 \\ -1 & 5 & 1\end{bmatrix}$ be asked symmetric matrix
$\therefore |A|=1(3+25)+2(2-5)+1(10+3) $
$=28-6+13=35$
$\therefore \operatorname{adj}(A)=\begin{bmatrix}28 & 7 & 7 \\3 & 2 & 7 \\
13 & -3 & 7\end{bmatrix} $
$\therefore A^{-1}=\frac{1}{35}\begin{bmatrix}28 & 7 & 7 \\3 & 2 & 7 \\13 & -3 & 7\end{bmatrix}$
Hence, inverse of a skew symmetric matrix is not skew symmetric matrix.