Question

If $A$ is matrix of size $n\times n$ such that $A^2+A+2I=0$, then

• A. $A$ is non-singular
• B. $A\ne 0$
• C. $|A|\ne 0$
• D. $A^{-1}=-\frac{1}{2}(A+I)$.

Answer 1

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

The correct options are
A $A^{-1}=-\frac{1}{2}(A+I)$.
B $|A|\ne 0$
C $A$ is non-singular
D $A\ne 0$

$A^2+A+2I=0$
$\Rightarrow A(A+I)=-2I$
$\Rightarrow |A(A+I)|=|-2I|\Rightarrow |A||A+I|=(-2{)}^n\ne 0$.
Thus, $|A|\ne 0$.
Also, $A\{-\frac{1}{2}(A+I\left)\right\}=I$
$\Rightarrow A^{-1}=-\frac{1}{2}(A+I)$.
Clearly $A\ne 0$ for otherwise $|A|=0$.
Hence, options A,B,C and D.