Up Main page

Definition

Let A be an n×n matrix and let k be a natural number. We define A0=In and Ak=AAk1.

From this definition, one sees that A1=AA0=AIn=A and A2=AA1=AA.

Example

[1102]3=[1102][1102]2=[1102]([1102][1102])=[1102][1304]=[1708]

If A is invertible, then Ak is also invertible and (Ak)1 is given by (A1)k. We define Ak to be (Ak)1.

Computing powers of matrices does not seem as easy as computing powers of numbers. However, there are cases when matrix powers are relatively easy to compute as you will see in some of the exercises.

Quick Quiz

Exercises

Compute each of the following matrix powers.

  1. [1211]3  

  2. [2003]6  

  3. [10002000i]2