Suppose A^k=0 for some integer k is greater than or equal to 1

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In summary, the expression "A^k=0" means that the matrix A raised to the power of k is equal to the zero matrix. This signifies that A is nilpotent and has important implications in linear algebra. A must be a square matrix with at least one zero eigenvalue for this expression to hold true. The value of k must be greater than or equal to 1, but the exact value will depend on the matrix A. This fact can be used to solve systems of equations, find eigenvalues and eigenvectors, and perform other computations involving matrices. It also has applications in data compression and control systems.
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Chris Rorres
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I was wondering if anyone could give me some hints on this

Suppose A^k=0 for some integer k is greater than or equal to 1. Prove that A is not invertible.
 
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Try a proof by contradiction. Assume it is invertible with inverse B. Then AB=1. (A^k)B=0B=0
But B=A^-1. So you can simplify (A^k)B=(A^k)(A^-1)=...
 
  • #3


Nice!
 

1. What does the expression "A^k=0" mean?

It means that the matrix A raised to the power of k is equal to the zero matrix.

2. What is the significance of having A^k=0?

This means that the matrix A is nilpotent, which implies that there exists a finite power k at which A becomes the zero matrix. This has important implications in linear algebra and can be used to solve systems of equations and other mathematical problems.

3. Can A be any type of matrix for A^k=0 to hold true?

No, A must be a square matrix for A^k=0 to hold true. Additionally, A must have at least one zero eigenvalue for this expression to be satisfied.

4. Is there a specific value of k that satisfies A^k=0?

Yes, k must be greater than or equal to 1 for A^k=0 to hold true. However, the exact value of k will depend on the size and properties of the matrix A.

5. How can we use the fact that A^k=0 to solve problems?

This fact can be used to simplify and solve systems of linear equations, find eigenvalues and eigenvectors, and perform other computations involving matrices. It can also be used in applications such as data compression and control systems.

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