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Math SL Portfolio: Matricies

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Introduction

Mathematics SL Portfolio Assignment 1

Title: Matrix Powers                                                                                                                           Type 1

1.Consider the matrix M =  Calculate Mnfor n = 2, 3, 4, 5, 10, 20, 50.

M²= * = = M³= * = = M⁴= * = = M⁵= * = = M¹⁰= * = = M²⁰= * = = M⁵⁰= * = = • To obtain the matrices above I multiplied (a*b+b*g), (a*f+b*h), (c*e+d*g), (c*f+d*h)as shown in the general formula seen below Describe in words any pattern you observe.

In the above matrices the pattern I observed is shown in relationship to the exponents and the numbers within the matrix. As the exponent increases consecutively the matrix is in turn multiplied by two.

Use this pattern to find a general expression for the matrix Mnin terms of n.

Middle

= * = = Determinant: 1296-784=512

P⁴= * = = Determinant: 18496-14400=4096

P⁵= * =  Determinant: 278784-246016=32768

= * = = Determinant: 400-256=144

= * = = Determinant: 12544-10816=1728

S⁴= * = = Determinant: 430336-409600=20736

S⁵= * = = Determinant: 15241216-14992384=248832

• To obtain the matrices seen above I used the matrix function on the TI 84 calculator. In order to obtain the discriminate in each problem I used the formula ad –bc such that .

Patterns Found:

The patterns portrayed in matrices Pⁿ and Sⁿ mainly correspond to the coefficient shown in the box and the determinant in each individual series. In the series of Pⁿ, as each exponent progress consecutively the coefficients associated with that particular matrix (in factored form) are multiplied by two (2, 4, 8, and 16). Also, each determinant found in the Pⁿ series is multiplied by 8 as each exponent progress consecutively (64, 512, 4096, 32768). In the Sⁿ

Conclusion = Results for k=(2), n=-(.5) = • According to the numbers substituted in the problem above, it can be inferred that the scope of the problem is all real numbers and fractions. I have found no limitations that hinder the use of k and n.

5. Explain why your results hold true in general.

Referring back to work shown in question three, my results for these statements hold true because of the amount of times the equations were carried out. Overall, the steadiness of the results for each of the problems ensures the validity of each statement. A wide array of aspects was used to prove these statements true ranging from negative integers to fractions. The patterns seen were also consistent with the results which reinforce the strength of my statements. Therefore, I conclude that my results hold true in general.

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

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