Question

Use (a) and (b) to show that if A is nilpotent and diagonalizable, then A is the 0 matrix. In other words, the only diagonalizable nilpotent matrix is 0.

A) True
B) False
C) The statement cannot be determined from the information given.
D) The question is not clear.

Answer 1

The student's question pertains to nilpotent and diagonalizable matrices, and the answer demonstrates that if matrix A is both, it must be the zero matrix. This is because a diagonalizable nilpotent matrix would have a diagonal form with all zero entries.

Step-by-step explanation:

The student has asked about the properties of nilpotent and diagonalizable matrices. Specifically, the question is to show that if a matrix A is both nilpotent and diagonalizable, then A is the zero matrix. To address this, we will use properties of matrices and definitions of nilpotent and diagonalizable matrices.

A nilpotent matrix is a square matrix A which satisfies Ak = 0 for some positive integer k, where 0 is the zero matrix. Diagonalizable means that the matrix A can be written as PDP-1 where D is a diagonal matrix, and P is an invertible matrix.

If A is diagonalizable and nilpotent, then its diagonal form D must also be nilpotent. However, a diagonal matrix is nilpotent if and only if all its diagonal entries are zero. Therefore, D and, consequently, A must be the zero matrix. Thus, the original statement that a nilpotent, diagonalizable matrix must be the zero matrix is true.

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