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Matrices SL Type 1

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Introduction

Mathematics Portfolio Type I

        The beginnings of matrices and determinants go back to the 2nd century B.C. although matrices can be seen back to the 4th century B.C., however, it was not until near the end of the 17th century that the ideas reappeared and developed further.  J.J. O'Connor and E.F. Robertson stated that the Babylonians studied problems which used matrices, as well as the Chinese using matrices early on.  All throughout history, the use of matrices has helped mankind progress.

        A matrix function such as X = image00.png and Y = image01.png can be used to figure out expressions.  By calculating X2, X3, X4; Y2, Y3, and Y4 the values of X and Y can be solved for.  image00.pngimage00.png = X2.  By following rules of multiplying matrices, this can be shown as image25.pngimage71.pngimage73.png = image75.png.  Using X2 we can conclude that X2 = image76.png or image12.png.  One can generalize a statement of a pattern that develops as the matrix goes on.  The expression is as follows, Xn = image02.png.  The number 2 in the matrix comes from when the product of Xn is solved for.  The value of 2n-1 is twice the value of Xn.  The variable n represents what power the matrix is to, such as n = 2, 3, 4.  We can now solve for the rest of the values of Xn.

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Middle

4 = image18.png + image59.png =  image74.png = 24image70.png.

The expression of (X+Y)n = 2nimage70.png fits and is proven by the examples above.

Let A = aX and B = bY wherea and b are constants.  Let us use different values of a and b to calculate the values of A2, A3, A4; B2, B3, B4.

a = 4 for A2

4image00.png∙ 4image00.png= image04.pngimage04.png =  image50.png

Solving the first example, we can create an expression that should work.  The expression for the value can be written as an2n-1X.  The 2n-1 comes from multiplying it with X, which is shown earlier on, Xn = image02.png.  The an comes from the constant of a which we solve the value of An and raise it to the n power.

Continuing using a = 4, A3 will now be solved for, using the expression.

A3

43∙23-1image00.png = 64∙22image00.png = 256image00.png = image31.png

A4

44∙24-1image00.png = 256∙23image00.png = 2048image00.png = image77.png

The same expression can be used for when B = bY, bn2n-1Y.  The 2n-1 comes from multiplying it with Y as shown earlier, Yn = image51.png.  bn comes from the constant of b we solve the value of Bn and raise it to the n power.  Since a and b need to be different constants, the value of b in this example will equal 5.

b =5

B2

52∙22-1image01.png = 25∙21image01.png = 50image01.png = image78.png

B3

53∙23-1image01.png = 125∙22image01.png = 500image01.png = image32.png

B4

54∙24-1image01.png = 625∙23image01.png = 5000image01.png = image79.png

Keeping in mind what A and B are, we can now find the expression for (

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Conclusion

image00.pngimage00.png.  The inner parts of the matrices must match, which the example follows since it’s a 2x2 ∙ 2x2.  The outer portions of the matrices are the resulting dimensions when multiplied.  For instances, you cannot multiply 2x1 ∙ 2x3 since the inner numbers in bold do not match.  The matrices used in all examples for finding general states have all been 2x2, which limits the general statements to only 2x2s.  The general statements would not work for a 3x3 or any others besides a 2x2.

        The general statement is Mn = an2n-1X +bn2n-1Y.  One would get to this general statement algebraically when multiplying A or B exponentially.  The 2 in the equation is twice as much as the square numbers and that is where the number 2 comes from in the general statement.  Since 2 receives less than the power n and this is where the section of n-1 arrives from in the equation Mn = an2n-1X +bn2n-1Y. When M = A + B, A = aX and B = bY are given earlier on in the paper from their expressions that were found by solving various problems.  An example would be;

a= 1 b =3, n = 2

M2 = 12∙21-1image00.png + 32∙21-1image01.png = 1∙1image00.png + 9∙1image01.png =

image00.png + image43.png = image44.png.

This is the algebraic step and method for solving the general statement of Mn = an2n-1X +bn2n-1Y

Works Cited

O'Connor, J.J. and Robertson. "Matrices and determinants." Matrices and determinants

Feb 1996. 13 Feb 2008 .

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