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

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A matrix A can be constructed such that A^3 = 0 while A^2 ≠ 0, typically by using a 3x3 upper triangular matrix with ones above the diagonal. The discussion highlights the challenge of finding such a matrix and the reasoning behind the structure of nilpotent matrices. Participants express uncertainty about how to approach the problem and suggest looking into relevant mathematical concepts or resources. The importance of understanding matrix multiplication and linear combinations is emphasized. Overall, the conversation revolves around the properties and construction of nilpotent matrices.
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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.
 
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NullSpace0 said:

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.
 
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?
 
NullSpace0 said:
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
 
NullSpace0 said:
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
 

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