If $A$ is a square matrix of order 3 and $A^2+A+2 I=0$, then
If $A$ is a square matrix of order 3 and $A^2+A+2 I=0$, then
A can not be a skew-symmetric matrix
$|A+I|=0$
$A$ is non singular and $A^{-1}=(A+I)^{-1}$
$|A||A+I|=2$
Solution
Given matrix equation $A^2+A+2 I=0$
$
\Rightarrow \quad A(A+I)=-2 I
$
$
\begin{array}{lc}
\Rightarrow & |A(A+I)|=|-2 I| \\
\Rightarrow & |A||A+I|=(-2)^3 \\
\Rightarrow & |A||(A+I)|=-8 \\
\Rightarrow & |A| \neq 0 \text { and }|A+I| \neq 0
\end{array}
$
and the determinant of skew-symmetric matrix. having odd order is zero. By here $|A| \neq 0$. So, $A$ can not be a skew-symmetric matrix