Question

i need to find the 100th power of the following 3x3 matrix......

2,0,0
0,0,-1
0,3,0

my question is can interchange row 2 and 3 before calculation. because that will make it diagonal and its easy to calculate the power of a diagonal matrix....

Answer 1

yes - work out which matrix you need to multiply your matrix by in order to swap rows 2 and 3.

Answer 2

then note what happens to that matrix when you square it

Answer 3

i.e. lets call your matrix A, then find a matrix B such that:

C = BA

and C contains A with rows 2 and 3 swapped

Answer 4

then note what \(B^2\) is equal to

Answer 5

and take advantage of that

Answer 6

do determinants

Answer 7

you mean in this case multiply with 1 0 0, 0 0 1, 0 1 0 right??

Answer 8

yes

Answer 9

so its alright to do that??? ii mean i was thinking it might change the property of original matrix A and so can not be performed

Answer 10

hang on...

Answer 11

I made a mistake up there, what you actually need to do is to find a matrix P such that:

P^-1 A P = D

where D is a diagonal matrix.

Answer 12

and then make use of:

A^n = (P D P^-1)(P D P^-1)(P D P^-1)...

Answer 13

ohhhh.....
i know that....and i was trying to avoid that long path

Answer 14

which simplifies to:

A^n = P D^n P^-1

Answer 15

because it has a structure very close to diagonal

Answer 16

so i was wondering whether a swqapping (if allowed) would save a huge time

Answer 17

no - because if you use my initial suggestion of:

C = BA

then you will get:

A = B^-1 C

so:

A^n = (B^-1 C)(B^-1 C)...

which doesn't really simplify