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If $A$ is a square matrix of order 3 and $A^2+A+2 I=0$, then
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Verified Answer
The correct answer is:
A can not be a skew-symmetric matrix
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.
$$
\Rightarrow \quad A(A+I)=-2 I
$$
$$
\begin{array}{lc}
\Rightarrow & |A(A+I)|=|-2 I| \\
\Rightarrow & |A||A+I|=(-2)^3 \\
\Rightarrow & |A||(A+I)|=-8 \\
\Rightarrow & |A| \neq 0 \text { and }|A+I| \neq 0
\end{array}
$$
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.
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