Q44 / 160 · AP EAMCET 2018 (22 Apr Shift 2) · Mathematics
If $A$ is a square matrix of order 3 and $A^2+A+2 I=0$, then
- Correct answerAA can not be a skew-symmetric matrix
- B$|A+I|=0$
- C$A$ is non singular and $A^{-1}=(A+I)^{-1}$
- D$|A||A+I|=2$
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.
Hence, the correct answer is option: 'A'.