Question

If $A$ is a square matrix such that $A^2+A+2I=O,$ and $I$ is an identity matrix of same order as that of $A$, then which of the following is/are true:

• A. $A$ is non-singular
• B. $A$ is symmetric
• C. $A$ can't be skew-symmetric
• D. $A^{-1}=-\frac{1}{2}(A+I)$

Answer 1

Answer: (A), (D)

$A^{-1}=-\frac{1}{2}(A+I)$

Given,
$A^2+A+2I=O\Rightarrow A^2+A=-2I\Rightarrow |A^2+A|=|-2I|$
$\Rightarrow |A||A+I|=(-2{)}^n\Rightarrow |A|\ne 0A^{-1}=-\frac{1}{2}(A+I)$