Need help with linear algebra pleease?

Question

An n x n matrix is said to be nipotent if for some natural number r, A^r=0

1) give an example of a non trivial nilpotent 2x2 matrix
2) prove that if an n x n matrix is invertible, then it is nilpotent

thanks

Answer 1

1)

[-4 -8]

[ 2 4 ]



Every matrix

[a b]

[c d]



where

(a + d = 0) & (a^2 + bc = 0)



is nilpotent.





2) It is not true, for example identity matrix is invertible and it is not nilpotent.



I^n = I

Answer 2

1) Try A =

[0 1]

[0 0].



2) You mean "if an n x n matrix is invertible, then it is not nilpotent."



Suppose A were nilpotent. Then A^r = 0 for some r in N.

Taking determinants yields |A^r| = |0|

==> |A|^r = 0

==> |A| = 0, and thus A is not invertible; this is a contradiction.



I hope this helps!

Answer 3

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