Proof: Nilpotent Matrix A & Real Matrices AB-BA=I

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In summary, a nilpotent matrix is a square matrix where all elements are 0 except for the main diagonal, which can be any real number. It also has the property of A^n = 0 for some positive integer n. A nilpotent matrix cannot be invertible due to its determinant being 0. Nilpotent matrices are significant in linear algebra as they represent linear transformations with limited range and have applications in solving systems of linear equations, differential equations, and Markov chains. The matrix AB-BA is related to nilpotent matrices as it will always equal the identity matrix I if A is nilpotent. A non-zero matrix cannot satisfy the equation AB-BA=I as the trace of AB-BA is
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kumarrukhaiyar
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1. A square matrix A is called nilpotent if a^k = 0 for some k > 0. Prove that if A is nilpotent, then I + A is invertible.

2. Show that the equation AB - BA = I has no solutions in n x n matrices with real entries.
 
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  • #2
Regarding 2, try to use the matrix trace function.
 
  • #3
1) Forget matrices. You want to find

(1+x)^{-1}

well, what is that as a power series?
 

1. What is a nilpotent matrix?

A nilpotent matrix is a square matrix in which all of its elements are 0, except for the ones on its main diagonal, which can be any real number. It is also defined as a matrix where A^n = 0, for some positive integer n.

2. Can a nilpotent matrix be invertible?

No, a nilpotent matrix cannot be invertible. This is because a nilpotent matrix has a determinant of 0, which means it is not possible to find an inverse for the matrix.

3. What is the significance of a nilpotent matrix in linear algebra?

Nilpotent matrices are important in linear algebra because they represent linear transformations that have a limited range. They can also be used to solve systems of linear equations and have applications in areas such as differential equations and Markov chains.

4. How is the matrix AB-BA related to nilpotent matrices?

The matrix AB-BA is related to nilpotent matrices because if A is a nilpotent matrix, then AB-BA will always equal the identity matrix I. This is known as the characteristic property of nilpotent matrices.

5. Can a non-zero matrix satisfy the equation AB-BA=I?

No, a non-zero matrix cannot satisfy the equation AB-BA=I. This is because the trace (sum of diagonal elements) of AB-BA is always 0, while the trace of I is non-zero. Therefore, the equation can only be satisfied when A and B are both nilpotent matrices.

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