Question

If $A$ is a square matrix of order $3$ such that $A^2+A+4I=0$, where $0$ is the zero matrix and $I$ is the unit matrix of order $3$, then

• A. $A$ is singular and $A+I$ is non-singular
• B. $A$ is non-singular and $A+I$ is non-singular
• C. $A$ is non-singular and $A+I$ is singular
• D. $A$ is singular and $A+I$ is singular
• E. $A$ is non-singular and $A-I$ is singular

Answer 1

The correct option is B $A$ is non-singular and $A+I$ is non-singular

Given,
$A^2+A+4I=0$
$\Rightarrow A^2+A=-4I$
$\Rightarrow A(A+I)=-4I$
$\Rightarrow |A(A+I)|=|-4I|$
$\Rightarrow |A||A+I|=|-4||I|$ (by property)
$\Rightarrow |A||A+I|=4\cdot 1(\because |I|=1)$
$\Rightarrow |A||A+I|=4\ne 0$.
So, both $A$ and $(A+I)$ are non-singular.