Matrix diagonalization - power of matrix question help

[Eagerissac]

Let A =

[10 5 -3]
[16 9 -5]
[52 28 -16]

Let x be a positive integer. If we calculate A^x =

[a b c]
[d e f]
[g h i]

what would the value of f be?

So using an exponential matrix calculator online, I got some of these values in the f position and tried to figure out a pattern:

x = 1, 2, 3, 4, 5, 6

Corresponding f values:

-5, -13, -29, -61, -125, -253

The formula I got to describe this pattern is f = (8) * (2^(x-1)) - (3) * (-1) but I'm told this is wrong. I thought it seemed to work. For example if x = 3, then f = (8) * (2^(3-1)) - (3) * (-1) = (8) * (4) - (3) * (-1) = -29 which seems to be correct as its the 3rd number in my sequence above.

I'm not sure why this is wrong or if I'm misinterpreting the question. I was wondering if I could get some help?

 

[Steven G]

Why is it wrong? Because there are infinitely many functions that satisfy the sequence -5, -13, -29, -61, -125, -253

For example if the next number was 1, then there is a 6th degree polynomial that contains those 7 points.
If the next number was 2, then there is a different 6th degree polynomial that contains those 7 points.
If the next number was 13.678945, then there is a different 6th degree polynomial that contains those 7 points.
....

There is also a 9th degree polynomial that contains those 6 points and a 12th degree polynomial that contains those 6 points.

You are studying algebra so you should know that the above is true.

Stop using software and solve the problem by hand by diagonalization.

 

[Steven G]

I guess that you are not up on your arithmetic as well.

I noticed that (8) * (2^(3-1)) - (3) * (-1) = (8) * (4) - (3) * (-1) = -29 is not correct. 32 - 3 is 29 so 32 - (-3) can't be 29 as there is only one number to subtract from 32 to get 29 and that number is 3!

Also why do have the formula as f = (8) * (2^(x-1)) - (3) * (-1) . Why not finish up by realizing what (3) * (-1) equals! f = (8) * (2^(x-1)) + (3).
-5, -13, -29, -61, -125, -253

Even if I use f = (8) * (2^(x-1)) - 3, none of the numbers are correct! For example if x=2, then f= 8*2-3 = 13, not -13. If x=3, then f=32-3 = 29, not -29.

Maybe you meant f = [(8) * (2^(x-1)) - (3)] * (-1) = -8(2^(x-1)) + 3