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

Matrices Portfolio

Extracts from this document...

Introduction

Matrix powers

A matrix is a rectangular array of numbers (or letters) arranged in rows and columns. These numbers (or letters) are known as entries. Entries can be added and multiplied, but also squared. The aim of this portfolio is to investigate squaring matrices.

When we square the matrix M = image00.pngimage00.png what we receive is a) image40.pngimage40.png = image51.pngimage51.png = image61.pngimage61.png.

Calculating the matrices for n = 3, 4, 5, 10, 20 and 50:

image71.pngimage71.png= image01.pngimage01.png

image13.pngimage13.png= image24.pngimage24.png

image35.pngimage35.png= image39.pngimage39.png

image41.pngimage41.png= image42.pngimage42.png

image43.pngimage43.png= image44.pngimage44.png

image45.pngimage45.png= image46.pngimage46.png

Examples shown above clearly indicate that while zero entries remain the same, non-zero entries change. Each entry is raised to a given power separately. Raising them to any power does not change the zero-entries.

...read more.

Middle

image60.png

These matrices are simplified to 2image56.pngimage56.png and 2image60.pngimage60.png to make it easier to notice any patterns. Calculating Pn and Sn 

...read more.

Conclusion

k and n it occurred, that for greater numbers the pattern was not true. In the case of, e.g. k = 1500 and n = 2, the pattern worked. Increasing n to 3, however, caused all the entries to be the same.

This was also checked for the matrices P and S.

image22.pngimage22.png = image23.pngimage23.png = image25.pngimage25.png = 65536image26.pngimage26.png

image27.pngimage27.png = image28.pngimage28.png = image29.pngimage29.png

image30.pngimage30.png = image31.pngimage31.png = image32.pngimage32.png = 4096image33.pngimage33.png

image34.pngimage34.png = image36.pngimage36.png = image37.pngimage37.png

The pattern works as long as the results are less than 10 billions. If they exceed this number, all the entries will be exactly the same.

image38.pngimage38.png 10 000 000 000

Therefore, this does not seem to be true for every number.

The results hold true in general because in real life situations so large numbers are not frequently used. For smaller numbers the pattern fits thoroughly.

...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. Math IA binomials portfolio. As we can see, a general trend emerges as we ...

    We can find an expression for Yn in a similar fashion. , where Un=a specific term a=first term r=common ratio (multiplier between the entries of the geometric sequence) n=the number of the specific term (with relation to the rest of the sequence). For Yn: When n=1, 2, 3, 4, ...

  2. Math Portfolio: trigonometry investigation (circle trig)

    As it can be seen in the diagram, all x and y values in Quadrant 3 are negative, though the r value remains positive as it is a radius of a circle, which cannot be negative. As all three formulas for calculating the sine, cosine and tangent values involve x and y, all are affected.

  1. Maths Project. Statistical Analysis of GCSE results at my secondary school summer 2010 ...

    12 We 9 40 f 40 11 We 10 34 m 34 10 We 9 58 f 58 9 Wh 10 46 m 46 8 Wh 9 46 f 46 7 Wi 11 52 f 52 6 Wi 10 40 m 40 5 Wi 10 28 m 28 4 Wo

  2. matrix power

    You then multiply the pairs together and the sum of the products will give a single number which is the first digit of the new matrix. Therefore the matrix equation being solved will look like: Another easier way of solving matrices powers is raise the power of the digits inside

  1. Math portfolio: Modeling a functional building The task is to design a roof ...

    = a= - , b= Therefore: a = , b = So the equation will be y=x2 + x --------(12) This is the equation for the curved roof structure of height "h" I will find the dimension of the cuboid of maximum volume in this curved roof structure.

  2. Math HL portfolio

    which is shown in the graph below: 3. Investigate your conjecture for any real value of a and any placement of the vertex. Refine your conjecture as necessary and prove it. Maintain the labelling convention used in parts 1 and 2 by having the intersections of the first line to be and , and the intersections with the second line to be and .

  1. Math 20 Portfolio: Matrix

    Considering the follow arrays of rectangular diagrams shown below: In each diagram, there are two copies of the triangular diagram, a black one and a white one. The original triangular diagrams from stage 1 to stage 8 now transformed into rectangles.

  2. Math IB portfolio assignment - MATRICES

    It should be noted that all of the elements remained positive. Table 1: Matrices for Xn and Yn n Xn Yn 5 6 7 8 9 10 Now that we have an idea of different patterns when X and Y are raised to an exponent ranging from 1 to 4,

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