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Math Portfolio - SL type 1 - matrix binomials

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Introduction

Math Portfolio SL Type 1:

Matrix Binomials

This portfolio deals with the investigation of matrix binomials. This portfolio will cover the basic knowledge of matrix in order to explain the methods and the procedure of arriving at the general statement. The task was to deduce a general formula for given matrices and to determine scope and/or limitation of this general statement. The general statement was obtained by using an algebraic method. All matrix equations were typed using the program MathType. It will test the validity of the general statement using GDC (GDC-TI-83). The scope and limitations of this general statement will also be discussed in this investigation.

Matrix is “a rectangular array of numbers arranged into a fixed number of rows (horizontal) and columns (vertical).” Each number in the array or the collection of numbers is called an element or entry of the matrix. The order of elements of the matrix is: image00.png.

Matrix has dimension of m (rows) x n (columns). The numbers m and n are the dimensions of the matrix and it plays significant role in the calculation of the matrix. For instance, the dimension of the following matrix is (2X2) since it has two rows and two columns:image01.png. For the addition of the matrices, corresponding elements need to be added.

...read more.

Middle

image56.png

The following expressions, Xn and Yn, were found by finding patterns when X and Y were powered. By examining the previous result, the elements in the matrix doubled as the number of power increased. It was also found that the end products of the matrix were the result of two to the power of one less than the power of X.

image57.png

To test this expression, ‘n’ was replaced with 3. Previously, when X was to the power of 3, all elements were 4.

Example:

image57.png

If image58.png,

image59.png

image60.png

image43.png

This expression was tested by replacing ‘n’ with 3. The result obtained from using the expression and the result obtained from previous section all had elements of 4. Therefore, the expression that was formed is correct.

In the case of Y matrix, it was little different due to the negative signs. The Y matrix had similar patterns as the X matrix except for the two negative signs in the second and the third element. Therefore, for Yn, negative signs should be add in the position of the second and the third element.

image61.png

To test this expression, ‘n’ was replaced with 3. Previously, when Y was to the power of 3, the first and the fourth elements were 4 and the second and the third were -4.

Example:

image61.png

If image58.png,

image63.png

image64.png

image52.png

This expression was tested by replacing ‘n’ with 3.

...read more.

Conclusion

n. It is that ‘n’ can’t be less than 0 and must be natural numbers because this matrix can’t have a negative value and will not work if it’s complex number or irrational number. The value of a and b should be natural numbers because the matrix will not work if it’s complex number or irrational number. Additionally, ‘a’ cannot equal to both ‘b’ and ‘n.’ If this happens, the general formula for Mn will not work. A limitation was found when using the GDC. Although GDC was useful in calculating numbers, it could not find a pattern and form a general formula. Additionally, it could not use algebraic coefficients like ‘a,’ ‘b,’ and ‘n’ to find the general statement.

To form this general statement, the Pascal’s Triangle was used to expand the algebraic equations. Pascal’s triangle is ‘a geometric arrangement of the binomial coefficients in a triangle.’ Since it is very complicated to multiply all those algebraic form of the equation, this triangle was used to ease the expansion of the algebraic form. The calculator was used for the simple mathematical calculations and to test the general statement. The program MathType was used to type in all forms of the matrices in the computer.

.


Bibliography:

"Math Forum: Ask Dr. Math FAQ: Pascal's Triangle." The Math Forum @ Drexel University. Web. 04 Feb.

2010. <http://www.mathforum.org/dr.math/faq/faq.pascal.triangle.html>.

MathType. Vers. 6.6a. MathType. Computer software.

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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