Powers of matrices

We define:

A matrix is nilpotent of order n if

A matrix is periodic of order n if

A matrix is idempotent if

Examples:

Example: calculate Aⁿ, where

Mathematical induction method

Mathematical induction is a method of mathematical proof typically used to establish a given statement for all natural numbers. It says:

Let P(n) a property that may be true or not for each natural number n. If:

- P(1) is true

- If P(k) is true → P(k+1) is true

Then P(n) is true

In the example:

Exercises:

1) Classify these matrices:

2) Calculate Aⁿ where

Solutions:

1) A is nilpotent of order 4; B is idempotent; C is periodic of order 2