Question

If $A$ is a matrix such that $A^2+A+2I=0,$ then which of the following is/are true?

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

Answer 1

Answer: (A), (C), (D)

The correct options are
A $A$ is non-singular

C $A$ cannot be skew-symmetric

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

Given, $A^2+A+2I=0$
$\Rightarrow A^2+A=-2I$
$\Rightarrow |A^2+A|=|-2I|$
$\Rightarrow |A||A+I|=(-2{)}^n$
$\Rightarrow |A|\ne 0$
Therefore, $A$ is nonsingular; hence, its inverse exists. Also, multiplying the given equation both sides with $A^{-1},$ we get $A^{-1}=-\frac{1}{2}(A+I)$