- #1

Drakkith

Staff Emeritus

Science Advisor

- 21,243

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## Homework Statement

Show that if

*##A^2##*is the zero matrix, then the only eigenvalue of

*##A##*is 0.

## Homework Equations

##Ax=λx##.

## The Attempt at a Solution

For

*##A^2##*to be the zero matrix it looks like: ##A^2 = AA=A[A_1, A_2, A_3, ...] = [a_{11}a_{11}+a_{12}a_{21}+a_{13}a_{31} + ... = 0, a_{11}a_{12}+a_{12}a_{22}+a_{13}a_{32} + ... = 0] = [0, 0, 0, ...]##

(Rinse and repeat for the next row)

The eigenvalue of a matrix is a scalar ##λ## such that ##Ax=λx##.

So here we have ##AA=λA##

It looks to me like ##A## could be an infinite number of matrices, and that ##AA## would only rarely, if ever, equal ##λA## for any nonzero ##λ##. But I'm not sure how to prove it.