Matrix Pattern?

[Sarusta]

We were given this bonus assignment from our HL math class, to find a pattern for the matrix A in terms of n, defined as:

A

[2 1]

[1 0]

That would work for any given matrix A^n

For example:

A^2

[5 2]

[2 1]

A^3

[12 5]

[5 2]

Where n = 2/3/etc, and the vales in the matrix are determined based on n.

Our entire math class could not find a solution to this, even our teacher. Any ideas?

Edited June 1, 2009 by Sarusta

[Toffu-san]

We were given this bonus assignment from our HL math class, to find a pattern for the matrix A in terms of n, defined as:

A

[2 1]

[1 0]

That would work for any given matrix A^n

For example:

A^2

[5 2]

[2 1]

A^3

[12 5]

[5 2]

Where n = 2/3/etc, and the vales in the matrix are determined based on n.

Our entire math class could not find a solution to this, even our teacher. Any ideas?

If look carefully the first five "n" you have this:

A:

[2 1]

[1 0]

A^2:

[5 2]

[2 1]

A^3:

[12 5]

[5 2]

A^4:

[29 12]

[12 5]

A^5:

[70 29]

[29 12]

If we rename the matrix we will see it better:

A^n:

[a_n b_n]

[c_n d_n]

First we have that: b_n=c_n

Then:

a_1:2

b_1:1

d_1:0

We can see that:

i_n=2*i_{n-1}+i_{n-2}

We can only apply this rule since n=3.

I wish to have helped you

EDITED: Now you have to solve the recursion and to prove by induction that it's true.

Edited June 1, 2009 by Toffu-san