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

Matrix Binomials Portfolio

Extracts from this document...

Introduction

Math SL Matrix Binomials Portfolio

        This portfolio will investigate the properties of matrix binomials in order to determine a general statement for Mn where n is a real number and an integer, and M is the 2image00.pngimage00.png matrix image28.pngimage28.png

Let:

X = image48.pngimage48.png                                 Y= image59.pngimage59.png

X2 = image70.pngimage70.png = image01.pngimage01.png                           X3 = image12.pngimage12.png = image24.pngimage24.png

                                       X4 = image34.pngimage34.png

Y2 = image37.pngimage37.png        Y3 = image38.pngimage38.png

                                       Y4 = image39.pngimage39.png

(X+Y) = image40.pngimage40.png

(X+Y)2 = image41.pngimage41.pngimage42.pngimage42.png                    (X+Y)3 = image43.pngimage43.png

                                     (X+Y)4 = image44.pngimage44.png

Expressions for Xn, Yn and (X+Y)n

Xn = image45.pngimage45.png            Yn = image46.pngimage46.png       (X+Y)n = image47.pngimage47.png

n > 0,  

let: W = any 2x2 matrix, image49.pngimage49.png

W-n = image50.pngimage50.png It is not possible to

...read more.

Middle

image57.pngimage58.pngimage58.png

B3 = image60.pngimage60.png

B4 =  image61.pngimage61.png

Therefore:

(A+B) = image62.pngimage62.png

(A+B)2 = image63.pngimage63.png

image64.pngimage64.png

image65.pngimage65.png

(A+B)3 = image66.pngimage66.png

image67.pngimage67.png

(A+B)4 = image68.pngimage68.png

image69.pngimage69.png

Expressions for An, Bn and (A+B)n

An = image71.pngimage71.png           Bn = image72.pngimage72.png

                         (A+B)n = image73.pngimage73.png

n > 0,  

let: W = any 2image74.pngimage74.png2 matrix, image49.pngimage49.png

W-n = image50.pngimage50.png It is not possible to divide an integer by a matrix, n < 0 does not exist

n≠0

For any matrix where n=0 Wn = I    W0 = image51.pngimage51.png

Let: M = image28.pngimage28.png, M = A+B and M2 = A2+B2

A = aX = image13.pngimage13.png

...read more.

Conclusion

n must also be greater than zero. It cannot equal zero because any matrix raised to the power of zero would equal I the identity matrix, and it cannot be less than zero, because that would be the same as one divided by the matrix raised to the power of n. Since it is not possible to divide an integer by a matrix, it is not possible to raise any matrix to a negative exponent and this n must be greater than zero. Therefore the expressions determined in this portfolio are valid for 2image36.pngimage36.pngmatrices, where the value of n is greater than zero

...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

    -xs���(r)SW�Z�1/2�Æ��_��^4�4K;ö-'�(c)�r�Ǻ>-�6�Æ-�!����X[ǤC�Zh'_�:(r)*3\>(c)@�Â����:�| ��S-���1/4(5�OL�I-���I��}��gv��6zZhZ'y:�Æ&�-�=���Ì����c�M\ɯx"�7_ 1/4_>��|b��I�u_]��_�!|{"k'�(c)x-H�=i��Q�wt��å.��;1/2�.�,4� 1/4m�[�9�o�|S_����>...���Ú]"~Òµ-��97��[K��G�4-1u�{+yl�|?k(c)���3/4(r)>ä[^...-� G��]�j3/4;Ҧ�1/4ms/��7�-:�i���%�U�,zcx"ö�x"�6Ú�ƨ4�>�Y�Ҵ-wRÓµ[m#V���Þ'�mO�����Â9�<Ii"x�M�#:�<�<?��#{� �h�_&"�(tm)l<w�[1/2I�3/4��������Lk�+� ��?�q� �s�-0�|�(�-�ox��-����D��Z����[�"�N��>=�uMj*=aoj-0꺵����� ]B[0 ���D�d�1/43/4%�61/2��?�=����\�\q���Z�]�|UK�"W�&"m�"����ޣr<EqcØ�o���Wz-� m��o���Iá��n�q�O�qx'��[� xw�6�A��"H���<���g��8:....����D���� [email protected]� �E7¿ï¿½ï¿½ï¿½ k>0�'�$��M1l�t�4] �����>(MS_��Æu�oK�M/B7���'j-0�*XH��~&�տ|?���<�[msL�}��{�:���M�...��m_Bï¿½Ö ï¿½u?�6W��u��1/2-(c)�wO���]��:�G��..."S6`ݯ�� M�xK��>(�<[� �w3/4=�t�'°Ç1/4%�Ç48�9�x�\�,���Y���k� �x"(r)�m�(tm)�...��O� �'��L�"�/�%"�G}WN�O�^xf���V���j�Y��|[(c)_�Ú��C�^�h��(r)"����cu&�-�����-�G������M�ҵ���z"- ���x��~4�1��Q���"�k;� ��W�����["Ù�s�'�o �_ĺ�...� =n]CE��1mu-���-�'�?� |"a'K�e��4�"�t1/4(mtÖk1�N�l�-+��F����D�,Λ�sE"X���u(r)��x"S�:x�MV_�'�L� �4z���;� �oay�j���Pmk��A x���&� ��Zw�.��n�/�6)(c)xO¶~0�th�,|Cq�$�M3/4��N��4�O ��s�v:�Yn�""���@�$�m1/4 �S:}�Äk�zL^ �] � �|"x�^"W��" �v�G�tK1/29|3�kWz��z��-��SY�� �;�a�...��<Q���ÊÑ°pF�`[email protected]<��[email protected]@P � {��<Y�g �/�N��"]��� �&�s�y�k���P � � �_�G�=sZ�"��z�\�A�������7�=�U} O�1/4'j�����&����N��Mv�Z]Keio3��~��Ia> �+)��S�v�3/4�g�ˢx�T"\�6��M�" �t��eÖ´"1l ��"+�i4�-�(�=�Q*�1/49k�i��<��G�?x�]-�'�MÇt��OZׯ�u��?���{�-�4�4�-RK'(�����:���k� [�Ú�'�t�ٿ�|M6���x��I1/4�Y��\�����8 �u�J�H�ӵ;����r�'[�[�?3/4��|{��5�z�"���Ä-Ϭ�'�IJ�jz�...u-R��CV�N�us��j1��֡�YϧCqginQËz/�~x3Å+�|o"Z�P���VSi�^���*�3/4"�w���>�(c)A��oJ�ִɵ2��[

  2. Maths Portfolio Matrix Binomials

    Example: Hence we can say that our general statement is: Testing the validity of my general statement, to do this we had to get the same results for both expressions: Assuming: a = 2, b = 4, n = 1 Assuming: a = -1, b = -3, n = 1

  1. Math IA - Matrix Binomials

    (We can also check this on a calculator). Now, using the expression, we find the value of Y5 to be the same (proving the accuracy of our expression): Y5=25-1 =24 =16 = We conclude that is indeed a valid expression. To find an expression for (X+Y)n, we must first determine the patterns by calculating the values of (X+Y), (X+Y)2, (X+Y)3, (X+Y)4.

  2. Math Portfolio: trigonometry investigation (circle trig)

    Likewise, when the value of y is divided by the value of r, a negative number is divided by a positive number resulting to a negative number. The value of x equals a negative number in the quadrant 3 and the value of r equals a positive number as mentioned beforehand.

  1. Math Portfolio Type II

    0 10000 10 58980 1 25000 11 60784 2 51250 12 59354 3 64703 13 60504 4 55573 14 59589 5 62953 15 60323 6 57375 16 59737 7 61892 17 60207 8 58378 18 59382 9 61218 19 60133 20 59893 The above tabulated data can be represented by

  2. Stellar Numbers math portfolio

    An expression for the stellar number at S7 , the 7th term is S7= S6+ (S6-S5) + 12 (where 12 is the value of the second differences). This was derived through the pattern noticed above where essentially the sum of each row is equivalent to the next Sn value, or

  1. Math 20 Portfolio: Matrix

    This general statement can also express the pattern of triangular geometric shapes as it poses the same pattern as the series of 100 consecutive positive numbers. Nevertheless, there is also an entirely new method of finding a general statement that represents the nth triangular number in terms of n that is enabled by the triangular geometric shape.

  2. Math Portfolio

    1300 1150 1060 970 900 850 800 780 740 710 680 660 Graph 1 This chart indicates the flow rate of the river against the time. The software also gives the r2 value of the graph that indicates the accuracy of the line best fit, in relation to the points that have been plotted.

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