• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Math SL Portfolio: Matricies

Extracts from this document...


Mathematics SL Portfolio Assignment 1

Title: Matrix Powers                                                                                                                           Type 1

1.Consider the matrix M =image00.png

image01.pngCalculate Mnfor n = 2, 3, 4, 5, 10, 20, 50.

M²=image10.png * image10.png= image27.png=image15.png






M⁵⁰=image17.png*image14.png=image20.png= image21.png

  • 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.

...read more.


=image04.png*image23.png= image26.png= image28.pngDeterminant: 1296-784=512

P⁴=image26.png*image23.png=image29.png=image30.pngDeterminant: 18496-14400=4096

P⁵=image29.png*image23.png= image31.pngimage32.pngDeterminant: 278784-246016=32768

=image24.png*image24.png=image06.png=image33.pngDeterminant: 400-256=144

=image06.png*image24.png=image34.png=image35.pngDeterminant: 12544-10816=1728

S⁴=image34.png*image24.png=image36.png=image37.pngDeterminant: 430336-409600=20736

S⁵=image36.png*image24.png= image38.png=image39.png

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 thatimage40.png.

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

...read more.



Results for k=(2), n=-(.5) image08.png=image09.png

  • 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.

...read more.

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

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. Extended Essay- Math

    ��@?`L3�ܰ, "���%�{Âp4��p\ 7�mp|~O��U@'P,(~"4J e��...rG bP�C�T5�"� u 5��F-���X4#�-��(c)� íF���+��6�z�1/2"�Â1\I��� �$br1%�:L+f�3����bY��XU�� �M�-�Va[�1/2�G��* �c�I�q"pT\.W�k��Ä�fq��$</^o�w�G��%�F|~ ?�� �" "]?B2�(�-�ExH�%l��Dm�#1"�E,#^$ _��H$':ÉL�$�'.'�HoH�4 C�x���41/24�h3/4��d�-ÙG. 7�o'_'��2��Ð��f�VÒ¶Ñ�~�#� ����K�+�"J��n'�@/BoHO�O����J��� ϰ�!��0C#�]�y �"B1��Qr(5"["F� �!�/�>�Z�A�Y&,"("9SS>S3��3...Y��(tm)9��'��4 �E"Å%��(�- -�ܬ����y�Y�X��8���� ���=a����n�-�~�1/2��% �C�Î#'�� �"'�&�/�!�+�Ϲ`. .{(r)T(r)(r)a(r)Un-nS�(�r�[Ü<,<z<!<�<=< 1/4�1/4:1/4�1/4�1/47y?�1���...�� �-�s�������p�hx)HT ,�\���+tA�0AXM8H�T��ð¨ï¿½ï¿½ï¿½'v'yQ6Qs�� �/��b�b�b�b�ű�j��U��"�D�D��CIXRE2X�J�'FJ]*B�Z�(c)4��3/4t���72,2V2�2�2�e...d�e���'�'S- ""��'��[�g�wɯ(H(�*T*<V$+�(f(v(.+I*�+�R�TfT�V> ܯ1/4(c)���rQeAUH�[���S5&5[��jC�u� �n�u �8�+iJk�j6j�k�j�k�j�h hS��jO���x��(tm)��ץ�V�3/4����"Ó���o�l gc�j�f�a�f�k"225:d4bL1v2(r)0~e"`hr�d�T�4մ� cfiv��(c)9���y��'...�E�Å%���e...�[+ "".k��ºï¿½...��M"M�.��|W�(r)-���Ѷ��v�v�v������q`t�rht��h�x�q�I�)�(c)�(tm)��ù�y��ȥ�e�U�5�3/4�[�[�;���1/2�}u���"g="=r=&<E="<���������7���"�'u����c�s�g�����_�ß����\�v@a�| v`Q�B�nPI�b�apE�r�Y�éµï¿½]��C��\�Z�����"�Ð�H�Ȥ�GQ'Q�Q���'�-b,c�b�X�Ø8&$9���&A'�2�{�s��$�����d��1/4�""s(c)�T��1/2�{��3/4I�O;� ����gf�d�f�f�g�B�-d�ef�粯+�;'3gf�� ���1�Oh-8}}0��H�b^y��!�C����K�-�=|�'�#�#GU��:�=ql���B��"�(tm)"뢶b3/4�C�_Ox��[�Tr�"X_:]fU�Q.T~��gEP�"J��-"\'�N(r)U�U���;u�4���?��(tm)<kz��Z���["P�3/4ֹ��9�s uu�u��#�O���4�644r5-1/2_������4�l��qQ���--�K�R��-1/2/O\�1/4�U���k��N�2�-j��'�-ÚÚ§;�:-uZt�wiv�^-�~3/4��"��=Ä���)7W{�z��f�1/2��n��z<`702h98t��;�wn iu�ո�yO�^�}��m��í"�����=T}�1�>�H�QÏ�X߸�����?�y�h�ib�(c)���I��ga�-�'<ß�|�yq�%��'W\��_��n(tm)V(tm)3/4�����[�*S33/43��3/4�9��3/4d�w(r)a^a3/4{�da�����>n,�~b�t���k�51/4�4"�1/41/2r� �-�_�3/4���(r)3/4��mc��w���j�w~����H���Y�)3/4Ùµe��b;|{;�C�� � �r-�Ý�a"���OC!� ��-�d[ZH�VF � ��0 � �Q+�|�f[...

  2. IB Mathematics Portfolio - Modeling the amount of a drug in the bloodstream

    The amount of drug in the bloodstream increases and decreases repeatedly. Method I started by entering the data given up to the sixth hour. At the sixth hour, I added 10 to the left over amount of drug in bloodstream.

  1. Mathematics (EE): Alhazen's Problem

    Initial approach: Now in order for us to have a rough idea of the range of possible solutions, we must consider several general cases and see what results we get. Consider Figure 4, here I have randomly chosen two points to be my locations for ball A and ball B.

  2. Math Portfolio: trigonometry investigation (circle trig)

    Based on this table we can see that there is relation between sin? and cos? from 0 to 90 degree. The example of sin30 equals to cos60 which the relation is that the theta is same. So the value of sin?

  1. Math Portfolio - SL type 1 - matrix binomials

    Therefore, Let A=aX and B=bY, where a and b are constants. There was a pattern found when A and B were each powered. For example, when A was to the power of 2, the result showed that both constant and the matrix were powered by 2 as well.

  2. IB Math Methods SL: Internal Assessment on Gold Medal Heights

    Hence, this hypothetical case has some weaknesses. According to the quintic function; the gold medal height in 1984 will be approximately 251 centimeters. On the other hand, the linear function gives a value of 239 centimeters for 1984. It appears that in this case, the linear function seems to be

  1. Math SL Fish Production IA

    This shows the rearranged years and the simplified equation of table 9. Rearranged Years (x) Substituting the x and y values 2 32a+16b+8c+4d+2e=16.1 3 243a+81b+27c+9d+3e=40.5 4 1024a+256b+64c+16d+4e=79.5 5 3125a+625b+125c+25d+5e=20.6 6 7776a+1296b+216c+36d+6e=63.3 These equations are rearranged into a matrix to solve the unknown coefficients: When the matrix is solved the coefficients

  2. Mathematic SL IA -Gold medal height (scored 16 out of 20)

    First of all, I will link all the points together to show a lined curve Figure 2 the curve of linking all data points of winning gold medal heights. The graph shows an upward sloping curve which is very irregular.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work