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**1. Matrix A^2 = 0, then A + I is nonsingular. Prove this or show a counter-example**

## Homework Equations

I suppose you must know Gauss-Jordan and matrix multiplication.

## The Attempt at a Solution

I first thought this was a false statement, so I tried to provide a counter-example. I tried with 2x2 matricies, both imaginary and real valued. For example I let A = [[1,1],[-1,-1]], which of course is A^2 = 0; however, A + I is nonsingular. With imaginary numbers I get the matrix of the familiar form [[a,-b], [b, a]], which is nonsingular (every matrix of that form is invertible, real valued or not).

So my thought now is that it is nonsingular. My issue is not so much the process of proving it, but what the insight might be as to why it would be nonsingular. How does the fact that A^2 = 0 help? What does it change about A's form in order to guarantee that A + I is nonsingular? Or does it not guarantee this and I just somehow missed a good counter-example.

-Ian