Matrix A s.t. A^3=0 but A^2 doesn't

[NullSpace0]

Homework Statement

Find a matrix, A, such that A³=0, but A²≠0.

Homework Equations

The Attempt at a Solution

I don't actually think this is possible. My only other thought is to take A²*A where this product equals zero, and then decompose A. But I don't know how to do that such that it is the "square root" of A.

 

[Dustinsfl]

Homework Statement

Find a matrix, A, such that A³=0, but A²≠0.

Homework Equations

The Attempt at a Solution

I don't actually think this is possible. My only other thought is to take A²*A where this product equals zero, and then decompose A. But I don't know how to do that such that it is the "square root" of A.

Try a 3x3 matrix with 1s above the diagonal.

 

[NullSpace0]

How would I have known to look for that?

From what I know about matrix multiplication, I know that the zeroes below the diagonal help when I think of multiplying by taking linear combinations of the columns, but how would I reason this out in general?

 

[Dustinsfl]

How would I have known to look for that?

From what I know about matrix multiplication, I know that the zeroes below the diagonal help when I think of multiplying by taking linear combinations of the columns, but how would I reason this out in general?

Let A be an n xn matrix for which A^k = 0 for some k > n. Prove that A^n = 0.

This theorem tells me to work on nxn matrices since I know I know I can construct one that is A^n = 0. So your case we needed A^3. Hence I picked a 3x3

 

[Ray Vickson]

How would I have known to look for that?

From what I know about matrix multiplication, I know that the zeroes below the diagonal help when I think of multiplying by taking linear combinations of the columns, but how would I reason this out in general?

Re "How would I have known to look for that?" I'll bet your book or notes have some relevant material. Alternatively, you could Google 'nilpotent matrix', but I guess you would need to know that is what such A are called.

RGV