# Idempotent and nilpotent matrix proofs

Not what you're looking for? Search our solutions OR ask your own Custom question.

Please show work.

A matrix pxp matrix A is idempotent if A^2=A...

Â© BrainMass Inc. brainmass.com December 15, 2022, 7:24 pm ad1c9bdddfhttps://brainmass.com/math/linear-algebra/idempotent-nilpotent-matrix-proofs-218166

#### Solution Preview

Please see the attachment

Problem #3

If is idempotent, then . Suppose is an eigenvalue of and is the corresponding eigenvector, then we have . Then we have

This implies that or .

So each eigenvalue of an idempotent matrix is either or .

If is a nilpotent matrix, then for some positive integer . Suppose is an eigenvalue and is the corresponding eigenvector, then we have . Then we have

This implies ...

#### Solution Summary

This provides examples of proving several statements about an idempotent and a nilpotent matrix.

$2.49