Question

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

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

Answer 1

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

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

Given : $A$ is a matrix of size $n\times n$ $\xi$ $A^2+A+2=0$

$\Rightarrow A^2+A+2=0$

$A^2+A=-2$

$A(A+I)=-2I$

$\left|A(A+I)\right|=|-2I|$

$\left|A\right||A+I|={(-2)}^n\ne 0$

Thus $\left|A\right|\ne 0$

Hence, the answer is $\left|A\right|\ne 0.$