# Thread: 1,1,1,1 Matrix general formula!!!

1. ## 1,1,1,1 Matrix general formula!!!

1) if matrix X=
|1 1 |
|1 1 |

find a general formula for X^n

2) if matrix Y=
|1 -1 |
|-1 1 |

find a general formula for Y^n

3) Now find a formula for (X+Y)^n

4)Let A=aX
and B=bY

-find general expressions for A^n, and B^n, using an algebraic method to explain how you arrived at your statement.

2. Hi

Evaluate $\displaystyle X^2$ and $\displaystyle X^3$, guess what is $\displaystyle X^n$ and show it using induction.

3. Hello,

Do it by induction.

$\displaystyle \begin{array}{cccc} X^2 & = & \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} & \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} \\ & = & \begin{pmatrix} 2 & 2 \\ 2 & 2 \end{pmatrix} & \end{array}$

Prove by induction that $\displaystyle X^n=\begin{pmatrix} ? & ? \\ ? & ? \end{pmatrix}$

Edited

4. Originally Posted by Moo
Hello,
Prove by induction that $\displaystyle X^n=\begin{pmatrix} n & n \\ n & n \end{pmatrix}$
Take a look at $\displaystyle X^3$, Moo

5. Originally Posted by daRitz
3) Now find a formula for (X+Y)^n
Hint: X+Y = I

For the induction hypothesis in the previous problem, $\displaystyle X^n = \begin{pmatrix} 2^{n-1} & 2^{n-1} \\ 2^{n-1} & 2^{n-1} \end{pmatrix}$

6. Originally Posted by flyingsquirrel
Take a look at $\displaystyle X^3$, Moo

Oh yeah, $\displaystyle 2^{n-1}$

7. does the formula X^n= 2^n-1 work?

8. No, it's $\displaystyle X^n=\begin{pmatrix} 2^{n-1} & 2^{n-1} \\ 2^{n-1} & 2^{n-1} \end{pmatrix}$

9. now can anyone help me with all the other problems??

10. For $\displaystyle Y$ the idea is exactly the same : evaluate its first powers, guess $\displaystyle Y^n$ and show it by induction.

11. Originally Posted by daRitz
4)Let A=aX
and B=bY
-find general expressions for A^n, and B^n, using an algebraic method to explain how you arrived at your statement.
$\displaystyle A^n = a^n X^n$
You already know $\displaystyle X^n$, so continue....

but im alittle confused, am i solving for A^n?

could you please give me an example by substituting some numbers in?

13. Originally Posted by daRitz
but im alittle confused, am i solving for A^n?

could you please give me an example by substituting some numbers in?
You know to compute X^n right?(Moo and flyingsquirrel helped you with that)

So multiply that with a^n.

You should get

$\displaystyle a^nX^n=a^n \begin{pmatrix} 2^{n-1} & 2^{n-1} \\ 2^{n-1} & 2^{n-1} \end{pmatrix} = \begin{pmatrix} a^n2^{n-1} & a^n2^{n-1} \\ a^n2^{n-1} & a^n2^{n-1} \end{pmatrix}$

14. thanks!
I'm now on the problem "find a general formula for (X+Y)^n

and like you said, its the identity matrix, and its looking as though it's the same thing as with finding X^n and Y^n, but im not sure how to set up the general formula, any thoughts?

15. Originally Posted by daRitz
thanks!
I'm now on the problem "find a general formula for (X+Y)^n

and like you said, its the identity matrix, and its looking as though it's the same thing as with finding X^n and Y^n, but im not sure how to set up the general formula, any thoughts?
You know why its called the identity?
It has a property.... You need not set up a general formula

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