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# Type 1 Portfolio: Matrix Binomials

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Introduction

## Type 1 Portfolio: Matrix Binomials

 Hudson Liao12/7/2008

I was given the expression X =  and Y = , where I calculate X2,X3,X4;Y2,Y3,Y4. Below I calculated X2 and made my way up to X4, where I also did the same with Y2 to Y4.

X2=

X3 or X2 * X1 =

X4 or X3 * X1 =

Y2 =

Y3 or Y2*Y1 =

Y4 or Y3+ * Y1 =

 Xn X1 = 1X = 20X X2 = 2X = 21X X3 = 4X = 22X X4 = 8X = 23X

Now I am going to find and expression for: [Xn, Yn, (X+Y)n], by inputting different ‘n’ values. By doing this I can find a correlation between each variable.

Expression: Xn = 2(n-1) X

This general statement was found by finding a relationship through values from X1 to X4. In the Xn table, a pattern begins to form from 1X, 2X, 4X and 8X. If we simplify these numbers by using a constant value such as 1X = 20X we can find a general statement for this expression.

 Yn Y1= 1Y = 20Y Y2= 2Y = 21Y Y3= 4Y = 22Y Y4= 8Y = 23Y

Expression: Yn = 2(n-1) Y

The same method to determine the general statement for the expression Xn = 2(n-1) X was also used for Yn = 2(n-1) Y.

 (X+Y)n 2I 4I 8I 16I

Middle

8Y

b=2

8Y

32Y

128Y

b=3

18Y

108Y

648Y

b=4

32Y

256Y

2048Y

By considering integer powers of A and B, find expression for An , Bn and (A+B)n

For the statement A= aX, I am going to determine a general formula by inputting different numbers for the constant ‘a’ and as well for the terms ‘n’.

An=(aX)n =

If I input a=1and n=2 into An=(aX)n, the resulting value would be 2X:

However if I continue to input a=1  and change the terms ‘n’ to 3 and 4a pattern begins to form:

By changing the terms n=2 up to n=4 ‘X’ increases each time from, 2X, 4X to 8X. The number of X’s that is being increased is resulted from this expression, “2n-1”. Therefore, we can convert the formula An=(aX)n to:

The same expression can be also used for the statement B=bY because both of the statements, A=aX and B=bY have the same pattern. The only difference between the two statements is that ‘X’ and ‘Y’ have different matrices. Therefore we just change, An  to Bn by:

If the same values that were inputted for An=aX to Bn=b

Conclusion

‘n’ cannot be a negative number.

In this second example I am going let ‘n’ equal to a positive integer and the constants a and b equal zero:

The limitation for the expression  is that ‘n’ cannot contain a negative exponent nor a decimal value or a fraction because if we multiply an exponent raised to a negative number it would make the value flip. However, both of the constants ‘a’ and ‘b’ can equal to any set of real numbers. Therefore the limitations and scope are:

Use an algebraic method to explain how you arrived at your general statement.

The general statement that came from  is   This general statement should equal to .

To prove that this general statement equals to , I am going to expand the equation  by using only variables:

 .=

Therefore the equation equals with

However, the equation would not work unless it is proven by the binomial theorem.

xn-kyk

=+2+

This calculations tells us that AB must equal to zero for this equation to work . As said before the only way the equation  works is because  equals to a zero matrix: .

In the end, the expression  can be substituted into a different equation where (aX)n+(bY)n can be replaced as .

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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