Provide an example for the following 3x3 matrix

[sigma]

A 3x3 nonzero matrix C such that C³=0 but C²\(\displaystyle \ \\\not =\\)0

I tried it with a bunch of different matrices and either I keep getting C² equal to 0 or just something else way off. Its easy to get C² to equal 0 but obviously you can't have that. I'm not sure what kind of approach to take towards a question like this other than just going through a bunch of trial and error calculations but I can't figure this one out.

 

[galactus]

I believe what you have here is a nilpotent matrix.

Here some some pointers to help you along in your quest:

a. Determinant of nilpotent matrix is 0

b. Eigenvalues of nilpotent matrix are 0

c. Nilpotent matrices are not diagonalizable

d. the trace is 0.

e. The characteristic polynomial is $\displaystyle {\lambda}^{2}$

 

[pka]

Try this one:
\(\displaystyle \left[ {\begin{array}{ccc}
0 & 0 & 1 \\
0 & 0 & 0 \\
0 & 1 & 0 \\
\end{array}} \right].\)

 

[sigma]

Thanks pka. How did you figure it was that matrix? I would have never guessed to use that matrix. I wish I can figure out how exactly what steps to go through for questions like this. It seems as though trial and error is the only thing that works, but for exams it would take too long, you have to know how to do it without going at it blind.

 

[pka]

How did you figure it was that matrix? What steps to go through for questions like this.

I could be flip and say that forty years of experience really helps. And in part that is certainly true. I realized that whoever wrote the question must have had a simple matrix in mind, one with zeros and ones. Then it is a matter of trying some configurations. Having a computer algebra system really helps.

 

[galactus]

One helpful tidbit about nilpotent matrices is if they are strictly

upper or lower triangular, they are nilpotent.

For instance, let's use a strictly upper tringular matrix.

That's a matrix in which all entries below the main diagonal are 0 and the

main diagonal itself is all 0.

\(\displaystyle \L\\A=\begin{bmatrix}0&2&2\\0&0&2\\0&0&0\end{bmatrix}\)

\(\displaystyle \L\\A^{3}=\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}\)

\(\displaystyle \L\\A^{2}=\begin{bmatrix}0&0&4\\0&0&0\\0&0&0\end{bamtrix}\)

Try a strictly lower triangular one.