• 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...


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.



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.


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)

    goes from -1 to 0. The value of ? increases from 0 to , sin ? goes from 0 to 1. The value of ? increases from to , sin? goes from 1 to 0. The value of ? increases from to , sin? goes from 0 to -1.

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

    Therefore width of the cuboid = 2v = 41.56 meters The value is same as in the case of 36 meters height structure hence I can say that the width of the largest cuboid does not change in the structure's height.

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

    Date : 10/06/2011 Name Entries English Language & Literature Mathematics FEMALE GENDER MALE GENDER Total score Alphabetic Position Number of Results 181 177 Ab 9 34 40 m 74 186 Ab 9 40 40 m 80 185 An 8 28 28 m 56 184 Ar 9 40 46 f 86

  1. matrix power

    For example, if, and In order to solve the matrix power above, we multiply the matrix by "n" number of times. Thus, if, and, then to solve we times the matrix "M" by two times. To multiply matrices, we take the rows of the left hand matrix and pair it with the column of the right hand matrix.

  2. Math HL portfolio

    In order to make the results more obvious, I created the following table where I listed all my results: Value for "a" 0.5 Value for "D" 2 1 1 2 0.5 2.5 0.4 ? 0.31833 This conjecture is only possible if the vertex is in the 1st quadrant and if the values of b and c are bounded to limitations.

  1. Matrix Powers Portfolio

    5 1024 n = 10 1048576 n = 20 1.099*1012 n = 50 7.88*1069 Again, there is an obvious pattern that can be seen between n, the exponent, and the determinant of the matrix. The determinant increases exponentially by the nth power, similarly to the top left and bottom right portions of the matrix.

  2. I.B. Maths portfolio type 1 Matrices

    = By calculating the values that were given in question number 1, I noticed a pattern that occurs from one value of n to the other. The 2 inside the matrix are always raised to the value of n in Mn.

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