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

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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.

Let X =   and Y =  Calculate X2, X3, X4; Y2, Y3, and Y4  

X2=  *  =                 Y2=   *  =  

X3=  *  =                 Y3=  *  =

X4=  *  =                 Y4=  *  =

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   =                                         Y7   =


X15 =                                 Y15 =


X20 =                                 Y20 =

 

X50 =            Y50 =

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  repeated.  =249

It should also be noted that whatever number X is raised to it is always 1 less than that of 2’s exponent

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The same applies for Yn except the elements b and c, these elements are negative. This is if matrix Y= .

 The following general statement can be made for Xn and thus, Yn:

Xn =  therefore, Yn =

To test the validity of this statement we will use different values for n:


From the results obtained from the tests outlined above we can conclude the following: X and Y do not have inverse matrices. This is because their determinant equals to 0(this was calculated using Microsoft Math)

The general statement’s limitations ...

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