Question

nilpotent matrix with some comman example​

Answer 1

Answer:

A square matrix A is called nilpotent if there is a non-negative integer k such that Ak is the zero matrix. The smallest such an integer k is called degree or index of A. The matrix A in the solution above gives an example of a 3×3 nilpotent matrix of degree 3.

Proof of (b).

If An=O, then by definition the matrix A is nilpotent. On the other hand, suppose A is nilpotent. Then by Part (a), the eigenvalues of A are all zero. Then by the same argument of the proof of part (a) (⇐), we have An=O.

The unit matrix is every n x n square matrix made up of all zeros except for the elements of the main diagonal that are all ones. For example: It is indicated as In where n representes the size of the unit matrix.

Answer 2

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