Matrices and Linear Systems

[IBVeryStressed]

God I suck at math

The square matrix X is such that X^3=0. Show that the inverse of the matrix (I-X) is I+X+X^2

[genepeer]

wow, took me a while to actually understand the question but now i get it. The basic idea is:

AA^-1=I

therefore, if (I-X) is inverse of I+X+X²

then (I-X)(I+X-X²) = I <- this is what you want to show.

so:

(I-X)(I+X-X²)=I²+IX+IX²-XI-X²-X³

=

I+X+X

2

-X-X

2

-X

3

=

I-X

3

=

I

because

X

3

=

0

, given in question.

Now that, we've showed (I-X)(I+X-X²) = I, we're done. In the exam, all you'll have to show is what i wrote in the quote.

Edited October 27, 2010 by Gene-Peer