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  4. If A is a square matrix.such that A3 = 0, then (I + A)-1 is

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Q. If $A$ is a square matrix.such that $A^3 = 0$, then $(I + A)^{-1}$ is

COMEDKCOMEDK 2005Matrices

Solution:

We have, $A^3 = 0$
$ \Rightarrow \, I + A^3 = I \, \Rightarrow \, (I +A)(I - A + A^2) = I$
$ \Rightarrow \, I - A + A^2 =(I+ A)^{-1}$
Hence, $(I+ A)^{-1}=I - A+ A^2$ .