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]

$⟹(A+I)(A^2-IA+I^2)=I$

$⟹(A+I)(A^2-A+I)=I$

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

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