Creating a Rule for A^n Matrix Power

[ronicencen]

Homework Statement

Given the matrix A= 1 2, make a rule for A^n
0 3

Homework Equations

A^2= 1 8 , so the first and last numbers are put to the power given (i tried other powers)
0 9

But then what about the 8?

Another one: A^5= 1 242
0 243

I need to get a general formula... it seems as though the 242 is the last number-1

But then if I have something like B= -1 1
-16 7
It goes all strange

B^2= -15 6
-96 33

Ok, if I make it C= a b
c d

Is this right for C^2: a^2+c ?
? d^2+c

By the way, the ? mean I don't know

 

[Gib Z]

Welcome to PF!

I don't understand why you introduced the new matrix B at all? The only reason they want you to find the pattern for A is because A has an easy to spot pattern! You've pretty much found it, except you just need to know the pattern for the 2nd entry on the first row.

 

[statdad]

Imagine starting as

$$\begin{align*} A &=\begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix} \\ A^2 & = \begin{bmatrix} 1 & 8 \\ 0 & 9 \end{bmatrix}\\ A^3 & = \begin{bmatrix} 1 & 26 \\ 0 & 27 \end{bmatrix}\\ \vdots & = \vdots \\ A^{10} & = \begin{bmatrix} 1 & 59048 \\ 0 & 59049 \end{bmatrix} \end{align*}$$

and continuing. What pattern do you see relating the entries in the first and second rows of column 2?

 

[ronicencen]

I see the pattern...but how would I make an equation?

 

[statdad]

An equation can be as simple as this one (totally made up, unrelated to your matrix problem)

$$a_1 = 2, a_2 = 3, \dots a_n = n+1$$

 

[HallsofIvy]

I see the pattern...but how would I make an equation?

I'm not sure you really "see the pattern" since if you did it would be easy to see the equation. Presumably you are able to see that the lower right corner number of Aⁿ, 3, 9, 27. 81, ... is 3ⁿ. How does the number directly above it 2, 8, 26, 80, ... relate to that?

 

[ronicencen]

It's -1?