1. 2x2 matrix diagonalisation formula.

Hi again overlords.

Looking for some clarification regarding this question. I know how to diagonalise a standard 2x2 matrix just following the process, however I'm having a problem with understanding the formula as provided by the solution to this question. Clearly, to obtain their formula they're just multiplying matrices. The thing confusing me a little is the 2^n+1 step, which I have highlighted. I can see they are just multiplying the two left most matrices, but I don't understand why this input is not 4^n. Furthermore, how is the input provided (2^n+1) even equivalent to 4^n? if you sub in some value for n greater than 1, then 4^n > 2^n+1.

2. Re: 2x2 matrix diagonalisation formula.

Hey nalo6451.

Hint - Try multiplying the first two matrices together and see if you get the same result as the next step.

Show your work here if you want a more detailed response.

3. Re: 2x2 matrix diagonalisation formula.

$2(2^n)= 2^{n+1}$ not $4^n$

4. Re: 2x2 matrix diagonalisation formula.

Oh MY BAD guys, I forgot that I was supposed to add the exponents as they have the same base (pray for me).

5. Re: 2x2 matrix diagonalisation formula.

I didn't want to ask again but I actually have a related problem on the following question. I've highlighted the exact multiplication step I'm referring to. Now, I know 1/2 is equivalent to 2^-1, so I'm trying to multiply 4^n with 2^n-1. I can't figure out why they've collected the powers (n-1) with the 4.

Thanks again friends.

6. Re: 2x2 matrix diagonalisation formula.

$\dfrac{1}{2}=\dfrac{2}{4}=2\cdot 4^{-1}$

$\dfrac{1}{2}\cdot 4^n = 2\cdot 4^{-1}\cdot 4^n=2\cdot 4^{n-1}$