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

Matrix Binomials. In this Math Internal Assessment we will be dealing with matrices.

Extracts from this document...


Math SL IA

Math Standard Internal Assessment

Matrix Binomials

In this Math Internal Assessment we will be dealing with matrices. A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are used for many things, such as; solving of systems and equations, linear programming, business inventories, Markov chains, strategies in games, economic modelling, graph theory, assignment problems, forestry and fisheries management, cubic spline interpolation, computer graphics, flight simulation, Computer Aided Tomography, Magnetic Resonance Imaging, Fractals, Chaos, Genetics, Cryptography, and the list goes on[1].

Let X = image00.pngimage00.png  and Y = image01.pngimage01.png Calculate X2, X3, X4; Y2, Y3, and Y4  

X2= image00.pngimage00.png * image00.pngimage00.png = image29.pngimage29.pngY2=  image01.pngimage01.png * image01.pngimage01.png =  image20.pngimage20.png

X3= image29.pngimage29.png * image00.pngimage00.png = image51.pngimage51.pngY3= image59.pngimage59.png * image01.pngimage01.png = image74.pngimage74.png

X4= image51.pngimage51.png * image00.pngimage00.png = image80.pngimage80.pngY4= image74.pngimage74.png * image01.pngimage01.png = image81.pngimage81.png

To find an expression for Xn and Yn we must test other values for n. These values were calculated using a Texas Instrument TI-84 Plus Graphic Calculator

X7   = image82.pngimage82.pngY7   = image83.pngimage83.png

X15 = image84.pngimage84.pngY15 = image85.pngimage85.png

X20 = image86.pngimage86.pngY20 = image87.pngimage87.png

X50 =image88.pngimage88.pngY50 =image89.pngimage89.png

It should be noted how the elements Xn are equal to that of the result of 2 raised to a number.

X2 has the element 2 repeated. 2 = 21

X3 has the element 4 repeated. 4 = 22

X4 has the element 8 repeated. 8 =23

X7 has the element 64 repeated. 64 = 26

X15 has the element 16384 repeated. 16384 = 214

X20 has the element 52488 repeated. 52488 = 219

X50 has the element image90.pngimage90.png repeated. image90.pngimage90.png =249

It should also be noted that whatever number X

...read more.


 = image03.pngimage03.png



=image05.pngimage05.png = 4(I)


(X + Y)n = (2I)n

(X + Y)3 = (2I)3

X3+ Y3 = (2I)3

image06.pngimage06.png + image07.pngimage07.png = 23image02.pngimage02.png

image08.pngimage08.png = image08.pngimage08.png



Another way to approach this question is to look at how matrix binomials differ from regular binomials. Let’s look at how binomial theorem works, if (a+b)² = a²+2ab+b² then in that case the following should take place (X+Y)2=X2+2XY+Y2. However, when matrix X is multiplied by matrix Y, their product becomes the zero matrix image11.pngimage11.png. Anything multiplied by 0 is 0, therefore the equation will simplify to (X+Y)2=X2+Y2. The same principle can be applied to n=3, (X+Y)3=X3+Y3. It should also be noted that when matrix X is added to Y the sum is the identity matrix, I= image12.pngimage12.png. The results of (X+Y)2 and (X+Y)3 are all multiples of the identity matrix. We can conclude once again, (X + Y)n = Xn +Yn=2n-1*X + 2n-1*Y= (2I)n

However there are still some limitations to this general formula. Seeing that (X + Y)n = Xn+ Y n , we know from the general expression for Xnor Y n

...read more.


a and b was less  than 0. It should be noted that since the forumula  Mn = An+Bn containsAnand Bn the same limitation exist as before where nimage13.pngimage13.png Z+. This conclusion on limitations should be followed up with a,b image78.pngimage78.png

In conclusion, by considering higher powers of the matrices of X and Y, patterns were observed and this led to the formulation of a general formula for (X + Y)nXn and Y n . These formulae were then tested for their validity by substituting several different numbers for n. It can also be concluded that he matrices A and B are actually multiples of the matrices X and Y. By testing different powers of the matrices A and B patterns were observed and this led to the formulation of a general formula for(A + B)n = An+ B n. This formula was then tested for its validity by substituting different numbers for a,b, and n. Then the matrix M was then introduced. By proving that  M = A+B, we were able to generate a general formula for Mn which was expressed in the form of aX and bY. The general formula for Mn has the following limitations: a and bimage79.pngimage79.pngand  nimage54.pngimage54.png Z+.

[1] (John Owen, Robert Haese, Sandra Haese, Mark Bruce 2004)

...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

    ��{w�I�"�[d� j�uץܤ\Y"�"�]i;��ǵ-J��y"��A����M�l5''���2�1/2�=P�s ; ���2��D�Ø|��](c)W�&��16�eÇ[2��N{l!��1/2 '=� #�i����Ic�!�i������^]�v� "u�%�...J��;mpx�Cy�n�#*��h]�"f��x��BÔCS�*���R�%~n"ßG_�lx�(r)�w�`��i=�;��ΦL"~2c� r(r)��g^'Ï���zbu1/2`3/4$�R<%�(r)� I����|��� �1/4'[�R�}��-�Y�r...7�"�L|)�cl��uج�æt���1/43��t ��5n\(tm)��YGN��DOw"lS�×Zg��@��#ñ¶ï¿½ï¿½ï¿½' >�R ...9"Ï°u�|Bf]����Û�j�pgI���"��ᨳ-�-UENr�]"�s 3/4j�5n���6�_��"��<� ��.d(c)�(c)V�`}"Hc�Q�J+-�Zn�(�����w��n�B���VG� -ilÖ^�l3�n�u"'%4v�...��I�6xtc�VNCL�s1/4WY(tm)L1/4' �-�B=�"�;m�(r)�}�!a�j'b��=��{���,SJ��9 D�Í\�ҳ�B[%;"��k�Bb�����`m(tm)�b�r ..._(tm)c<�W�G�-Óv�a[�caW�"��(i�(r)��-�G93o/�}� !2pU�5"Jag3�V����U-zC�׭mv2n.�'���(⢶S�p'"wÊ¡j"�Ó�R���[�:W�Ua�m�a�Jm}�v�O-�å¶ï¿½k�NE�J<��z�\^bJR;��Æ��n�/a�O0wo��d-(Ö|�"W���-2�Ì��"�{G�U��i(c)�t��r��i��e&�l-(tm)��18���q�(c)Û�� ��]z��͸U> �'��(r)Z �];�(tm)P��Xl�8��*�!��71/2�nrJ�U�!IÏ�F^(r)(tm)R~�-[[...CB�|Sv��C'YU�j��bg""Ø�����0Q�P��lc�Ç-n�(]�j���Z�"b��'Ú�F�e�-�'Ĥ(tm)2-���"5�ĵT�7A)(�����(c)�j�WC�;(r)Ë1/2�KD +�+V-�@g��|�5'�|�/��6�)�*E'��(tm)^Z����5�]�c_c�i*(c)f"q�r�\O6V"�V��V��J�Lî²h@-K�R!$:m�F�:���\�'�L�v��Y��ϳ|R�6�TTߦ�P"��#2k=<H$Bk=d�m�I,F=��C�1ËC"1/4��)����6��d��-�۬`�è���i,�|�iHh*Li�� "�IW�D���-�'1ҳ��XHB����@��u�]����O��i�pv� <M��1/2��@ W Í�Z� �38w��(tm)�2�c�"uҥ�@3/4�@3/49e3�|�.�|���$"<�&�n�z �|)���"�sr#�xl.U�m�Z�Y_��ZK<�1/2-/��c�,3��(�ʲ��wÅ��1/4��)S�W��Lv��fw1/2�^rI)��Y� �gJ"�^`��("�E1�-G"U�L��j����Ma^��A%�l��p"-�R3/4oO� 3/4�1/4+0�%��(�!�;�nz�- �Mg�"(tm)�J�)b��KQ��� ���"�(tm)QKP��IqtLÇ¢L��*�(c)#m��8�hz�A_{|���-��+ �k�"(r)��OQ}����bTM9)�� �4���*��(r)q�z�(c)Zq� �"8��!-q"z%-�p4�cN������|�����n�7Ç��-t3�2� _x��!���{(c)�'[iZ+f�...�\�Y"�52R�k;F�][��(tm)ÈC�Tpg{�%����eg�ԨVzǸ �Jy��c1/2D*_�'��^l!Õ�5Y����=e�idgM�'�J����I��c��xo<�w��"n�(c)i:��R[�L(r)08ØF�k�ҦÑwFUq�j���f�Íg��z!%VS �9[t:Q1�u��iR�(tm)jL^�6�;(CO�ë�p31/4�f�4��� � @���(x����� c'��ZR��Rs4'�K�i!#�D'�y$��'I1/4�%'#G�1�| ��f�"4/�e9vi�o-�nhk7�6�(c)���...6����1/4��I�_�'Ö�L�k7�L�(tm)-rÔ���)���*"���Q..."�G���U�/-'���N(c)'$*�u(Y�-rdy�^ƦZvͱ�1�s�o1|H�Q���\�^Us\T�6JBF�jc:��|���KL��w�|�s�xZb' :���Qy�cT2��J�$�����@��S�2�I�1/4( X@Z��(r)�<��'� $x@M-K�-z���%ÑW�3/4 xm(r)6�c��Ä��"n�z�92�~s#{� �h�u�)�I(,6�9��)"�"���Ұ� @�...�(r)����T���x\�j-�"�(tm)1/2 �S�]>�bkR�U[�Y���T�i��p4R8"�8Â�~�~/�S{x)��s��|��{"q��-�9?��#!�&��YL �g-)�?'"q�sZM�"q^Ud>E �.4��0�%���T�&�'����Í_h��J2��'"MÑ��1/2&H3�'5�[ c H4-v@.��3/4��=1W�Uz7����1/4�A�~�5�-��O�7�Ý^�-�`R�t{�x*�0��'1/4�l4�071/4=wWdFB�1/4�d���l�@�.+KI�*r�"_S�8O"��K��C�"C~�T�...�Ik���t3��Å�kY�(� 2 �3/4)6E��_L$�[�'u ."

  2. Math Studies I.A

    During the interpretation of results the weak correlation was partially explained by factors such as war, food shortage, economic crisis, epidemic of diseases, and so on. I addition, it could well be that there is a non linear correlation instead.

  1. Math IA - Logan's Logo

    Therefore, to find the value of c, I must first determine the center line of my curve - the middle line of the total height. In fact, we already determined this when we found variable a. (Recall that we divided the total height of the curve by 2 to find the amplitude).

  2. Maths Internal Assessment -triangular and stellar numbers

    -3 Stellar, layer 3 Sn = pn (n-1) + 1 Sn = ((-3)x(3))(3-1) + 1 Sn = (-9 ) x (2) + 1 Sn = -18 + 1 Sn = -17 = drawing -17 dots is impossible. Question 10: Explain how you arrived at the general statement.

  1. Math Studies - IA

    square regression line, the greater the European win in the Majors is the smaller their victory in the Ryder Cup is. Hence there is a negative correlation between the two variables. In other words, the better the US performs in the majors (going toward zero on the x-axis, since any value below on represents a US win)

  2. Mathematics Higher Level Internal Assessment Investigating the Sin Curve

    In general it is observed that in the equation the period of the curve is given by . By changing the value of you can change the period of the graph. When you increase the the period of the graph decreases and when you decrease the the period of the graph increases.

  1. MATRIX BINOMIALS. In this investigation, we will identify a general statement by examining the ...

    Thus, we know that M�=A�+B� and , therefore Here, the general statement that expresses in terms of aX and bY can

  2. Math IA Type 1 In this task I will investigate the patterns in the ...

    Some general conclusion that can be drawn from the analysis of changing the slopes of lines. * A change in the y-intercept of the intersecting lines has no affect on the conjecture * Changing the slope of the lines Therefore I have tested lines with different slopes and different y-intercepts

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