• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month
Page
1. 1
1
2. 2
2
3. 3
3
4. 4
4
5. 5
5
6. 6
6
7. 7
7
8. 8
8
9. 9
9

# In this portfolio,my assignment is to deal with matrix binomials,and to investigate them.

Extracts from this document...

Introduction

PORTFOLIO TYPE I – MATHEMATICAL INVESTIGATION

-MATRIX BINOMIALS-

Student: Domazet Neven SL

In this portfolio,my assignment is to deal with matrix binomials,and to investigate them.

At first,my initial matrices are as follows:

and

Now,my task is to find and calculate following matrices: : X2, X3, X4; Y2,Y3 and Y4

So,since  and  we will calculate that as

We know that we can express multiplication of matrices as ,and it goes like

,so we get that , thus.  Or 2* =2* X

As we now know this,it is easy to calculate ,because or 2*X*X,which is 2*2*X or it is equal to 22*X and equal to .

is defined as X3*X=22 *X*X=22 *2*X=23 *X and our solution is =

Procedure for Y is completely the same as for X,and we do not have to go in details,so to calculate the products of , and , we will use the TI-84 Plus Texas Instruments Graphic Calculator and obtain the results in the following form ( [B] equals Y)

The results are as follows: ;  and .

Middle

This proves that  and since we know that  and   this means that . Now we can calculate

So we can now see an expression for matrix as  and substituting  we can find a general form X also same goes for Y as  Y

Now when we have found  and   as expressions our next task is to find ,and it is quite simple:

Let us first sum up X+Y matrices and it goes like

X+Y= or 2*  and it is now obvious that this is in form of 2*E,as E stands for

Now when we now this it goes like

=(2*E)=2*E=2*= because E=E

Following these general expressions, we may continue to investigate the pattern in the following equalities  and  , when and  are constants and they have different values.

It is very important to mention that X*Y= and it is obvious that X*Y=0 and 0 stands for zero matrix

Now we can calculate =(aX)=a*X*X=2 *a*X

= *A=2 *a*X*(aX)=* X2=  * a **X

= *A=**X

Conclusion

Similarly we can prove first that M2=+ and we see that this is   and when we simplified this like

2* +2  and we now see that M2=+,it  is completely the same for numbers,as we have shown for M=A+B

There is also other way to prove this,as we now that M2 =(A+B)*(A-B),and we earlier proved that A*B=0(zero matrix),now it goes easily like

M2 =(A+B)*(A-B)= +A*B+B*A+=+0+0+=+

We also already proved that =

And now if  Mn =(A+B) n  we see that

Mn =

By taking for example a=2           b=-3           n=2

==*=

=2*(+)=2*=

Through my work,I have shown that  and  belong to the set of rational number  and belongs to set of natural numbers 0.

I have also shown that  and  when multiplied give a zero matrix: .and that A*B gives zero matrix as an in example with

in which like  and XY gives zero so that

Because we know that M=A+B than referring to the law of exponentiation we see that

Mn =

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

# Related International Baccalaureate Maths essays

1. ## Mathematics (EE): Alhazen's Problem

Please note that this essay (and the solution to the focus question) is narrowed down to emphasize the algebraic solution to Alhazen's Problem - however in the conclusion, other methods are briefly discussed. Pre-examination of the problem: The great difficulty with this investigation lies within two concepts.

2. ## Math Portfolio: trigonometry investigation (circle trig)

When we put a random angle from quadrant 4, the range of 270<?<360, 336� in trial to verify the conjecture, the value of cos turn out to be positive while the values of sin and tan turn out to be negative.

1. ## Math Portfolio - SL type 1 - matrix binomials

Let and By calculating powers of X and Y, a definite relationship was found between the power of the matrix and the elements. When X2, all the elements in the matrix were 2; all the elements were 4 when X3; all the elements were 8 when X4.

2. ## Stellar Numbers. In this task geometric shapes which lead to special numbers ...

a graph was plotted to demonstrate how it expanded Now, a p value of 6 will be used based on a regular hexagon: Stage Number Number of Dots Notes and observations 6S0 1 None 6S1 7 Adding 6 to previous 6S2 19 Adding 6x2 to previous 6S3 37 Adding 6x3

1. ## Matrix Binomials IA

n = is true and consistent with all values of n. 2. Consider A = aX and B = bY, where a and b are constants. In order to find the general expressions of An, Bn and (A + B)n, different values of a and b are used to calculate A2, A3, A4 and B2, B3, B4 below.

2. ## Math IA - Matrix Binomials

For Yn: When n=1, 2, 3, 4, ... (integer powers increase), then the corresponding elements of each matrix are: 1, 2, 4, 8, ... These terms represent the pattern between the scalar values multiplied to Y= to achieve an end product of Yn. Thus, we can now deduce the geometric sequence of these scalar values using the general

1. ## Stellar Numbers Investigation Portfolio.

- (Tn) Progression 1 1 2 1 2 3 3 1 + 2 3 6 4 1 + 2 + 3 4 10 5 1 + 2 +3 +4 5 15 6 1 + 2 + 3 +4 +5 6 21 7 1 + 2 + 3 + 4

2. ## The purpose of this investigation is to explore the various properties and concepts of ...

Attempted communication of emerging mathematical ideas and reasoning. Limited attempt to use appropriate notation, representations, or terminology, and with limited accuracy. ________________ FOLIO TASK: MATRIX CRYPTOGRAPHY Introduction Many communications that are transmitted between and within countries need to be secure. Coding the messages can provide that security. One method of encoding involves using an alpha numeric code that is then encrypted by matrix multiplication.

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