Find A^100

Question

Consider the Matrix A=                                                                                                                                                [ 1 -2 8 ]

 [ 0 -1 0 ]
 [ 0 0 -1]

or A = ( 1,0,0) , (-2,-1,0) , (8,0,-1)


(a) Find A^100

Answer 1

There are  2 ways to solve problems like that.

 

One is to use an online matrix calculator. There are quite a few. And the online matrix calculator explains steps you need to take to multiply matrices.

 

The other way (more mathematical in my opinion) is to get the matrices in the problem to the second power any way you can. And then to the third. And you will see that A^2=I₃ as in  identity matrix. So, A^3=A and so on. A^100=I_3.

 

It works for both matrixes in the problem.

 

I answered assuming that you know how to multiply matrices. Please let me know if you need and explanation on multiplication itself. 

 

 

Answer 2

Hi Rola,

 

I just calculated A*A, or A² and find that this is the identity I = {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}.

 

We can write A¹⁰⁰ = (A²)⁵⁰ = I⁵⁰ = I 

 

So, A¹⁰⁰ is just the 3 X 3 identity matrix I.