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)

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

$A^2+A+2I=0$

$A^2+A=-2I$

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

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

As $A^{-1}$ exists, $|A|\ne 0$ ,hence it is a non-singular matrix

Here $A=-A^T$ hence it is not skew symmetric.