Let A be a nonzero square n x n matrix. Is it possible that a positive integer k

Question

Let A be a nonzero square n x n matrix. Is it possible that a positive integer k exists such that Ak = 0? For example, find A^3 for the matrix

A = (3x3 matrix) row1: 0 1 2 row 2: 0 0 1 row 3: 0 0 0

A square matrix A is said to be nilpotent of index k if A doesnt equal 0, A^2 doesnt equal 0, . . . , A^k-1 doesnt equal 0, but Ak = 0. In this project you will explore the world of nilpotent matrices.

What is the index of the nilpotent matrix A?

Use a graphing utility or computer software program to determine which

of the matrices below are nilpotent and to find their indices.

(a) 2x2 matrix; row 1: 0 1 row 2: 0 0

(b) 2x2 matrix; row 1: 1 0 row 2: 0 1

(c) 2x2 matrix; row 1: 0 0 row 2: 1 0

(d) 2x2 matrix; row 1: 1 0 row 2: 1 0

(e) 3x3 matrix; row 1: 0 0 1 row 2: 0 0 0 row 3: 0 0 0

(f) 3x3 matrix; row 1: 0 0 0 row 2: 1 0 0 row 3: 1 1 0

3. Find 3 x 3 nilpotent matrices of indices 2 and 3.

4. Find 4 x 4 nilpotent matrices of indices 2, 3, and 4.

5. Find a nilpotent matrix of index 5.

6. Are nilpotent matrices invertible? Prove your answer.

7. If A is nilpotent, what can you say about A^T? Prove your answer. 8. If A is nilpotent, show that I - A is invertible.

Explanation / Answer

8) A is nilpotent with index k. (I-A)(I+A+.........+A^(k-1))=I

Hence (I-A)is invertible

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