Question

If $A$ is a non zero square matrix of order $n$ with $det(I+A)\ne 0$, and $A^3=0$, where $I,O$ are unit and null matrices of order $n\times n$ respectively, then ${(I+A)}^{-1}=$

• A. $I-A+A^2$
• B. $I+A+A^2$
• C. $I+A^2$
• D. $I+A$

Answer 1

The correct option is A $I-A+A^2$

$det(I+A)\ne 0$

$A^3=0$ where $0$ is null matrix, $I$ is the identity matrix

$A^3+I=I$ [adding $I$ on both sides]

$(A+I)(A^2-IA+I^2)=I$ [by the formula of $a^3+b^3$]

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

$(I+A)(I+A{)}^{-1}=I$ [by the rule of inverse matrix]

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

Ans: $I-A+A^2$