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Matrix Powers Portfolio

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Introduction

Colin Wick                  3/12/09

Period 7                Pre Calculus

Matrix Powers: Type I

        Matrices are useful mathematical tools that help us to interpret, represent, and ultimately understand information.  By comparing matrices in their original form, , we can observe specific patterns that are helpful in interpreting information demonstrated using matrices.  But, in some specific cases, the determinant of said matrices must be calculated in order to see a specific correlation.  This investigation analyzes relationships between both matrices and determinates of differing powers.  It further presents a general rule that can be used to calculate a specific pattern for any power.image00.pngimage01.png

        When considering the matrix: , an expression for the matrix of Mn in terms of n can be found by simply formalizing an observed pattern of the matrix Mn where n=1,2,3,4,5,6,7,8,9,10,20,50.  After plugging the matrix M into a graphing calculator, M to the previously stated powers can be easily calculated using the “^” key.  After making these calculations, the results obtained can be observed below in Table 1.1.

Table 1.1

Power

Matrix

n = 1

image12.pngimage13.png

n = 2

image14.pngimage15.png

n = 3

image16.pngimage17.png

n = 4

image18.pngimage19.png

n = 5

image02.pngimage03.png

n = 10

image04.pngimage05.png

n = 20

image06.pngimage07.png

n = 50

image08.pngimage09.png

An obvious pattern can be seen above, in which as the power n increases, so do the numbers in the top left portion (a) and lower right portion (d) of the matrix, while the other two values stay constant at zero.  In fact, the numbers in the top left and lower right increase exponentially.

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Middle

image02.pngimage03.png

1024

n = 10

image04.pngimage05.png

1048576

n = 20

image06.pngimage07.png

1.099*1012

n = 50

image08.pngimage09.png

7.88*1069

Again, there is an obvious pattern that can be seen between n, the exponent, and the determinant of the matrix.  The determinant increases exponentially by the nth power, similarly to the top left and bottom right portions of the matrix.  But, instead of having a base of two like the matrix, the determinant of the matrix will have a base of four.  Therefore, the determinant of Mn can be represented with the equation Det(Mn)=4n.

        Although the matrix M shows very obvious patterns when it is exponentially increased, if the zeroes in the matrix are replaced by numbers greater than zero, the patterns observed become far more complex and scale factors must be utilized in order to simplify them.  Consider the matrix S=       .  Because the matrix itself is so complex, there will be no evident pattern unless the matrix is factored to a point where a relationship can be observed and formalized into a general equation.  First, Sn must be found where n=1,2,3,4,5,10,20,50 using a calculator.  Then, a correlation between the scale factor and n must be found, in which the pattern observed in the matrix of S1, is observed in all matrices of Sn.  Therefore, the scale factor for the first matrix must be one.  In this case, the equation for the scale factor in terms of n is 2n-1, which makes sense because the numbers in the upper-left/lower-right and upper-right/lower-left have a difference of two.  A chart representing the original matrix, scale factor, and factored matrix can be seen below in Table 2.1.image20.png

Table 2.1 (continues on page 4)

Power

Matrix

Scale Factor

Factored Matrix

n = 1

image21.png

1

1image21.png

n = 2

image22.png

2

2image23.png

n = 3

image24.png

4

4image25.png

n = 4

image26.png

8

8image27.png

n = 5

image28.png

16

16image29.png

n = 10

image30.png

512

512image31.png

n = 20

image32.png

524288

524288image33.png

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Conclusion

image36.png

Table 3.1

K

N

image37.png

Factored Matrix

Determinant

-6

-2

Err. Domain

image38.png

0.001736

-6

0

image39.png

image40.pngimage41.png

1

0

3

image42.png

image43.png

0

image44.png

3

image45.png

image46.png

4096

Although the calculator was not able to verify the expression with a negative two, a similar pattern is seen in which the top-left/bottom-right values and bottom-left/top-right values have a difference of two.

        In conclusion, the relationship between n and both the determinant and matrix to the power of n, is possible to find.  But, the relationship between n and the determinant of the matrix to the power of n is far simpler and therefore easier to recognize.  Therefore, determinants are helpful in visualizing relationships between matrices that otherwise would seem completely divergent of one another.  Furthermore, using the equation for the derivative ((4k)n=det(Mn)) to represent patterns is far less confusing than attempting to show patterns through the equation for the factored matrix to the power of n (image47.png).  But, despite the obvious benefits of using one equation to show patterns over the use of the other, both generalized equations make the calculation of matrix powers and their derivatives without a calculator extremely efficient and nearly effortless.

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