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

# Math's Portfolio SL Type 1 &quot;Matrix Powers&quot;

Extracts from this document...

Introduction

Newton College

Math’s Portfolio

SL Type 1

“Matrix Powers”

Mauro Gelmi

IB 2

March 2009

Introduction

Matrices are tables of numbers or any algebraic quantities that can be added or multiplied in a specific arrangement. A matrix is a block of numbers that consist of columns and rows used to present raw data, store information or to perform certain mathematical operations

The aim of this portfolio is to find a general trend in different sets of matrices, therefore find a general formula that applies for all of the matrices. The pattern will then be explained and tested to see if applies correctly to all the sets of matrices.

Method

To calculate Mnfor n = 2, 3, 4, 5, 10, 20, 50 I used a GDC.

Determinants are defined as ad-bc in a 2 2 matrix, and is denoted by det A = | A |. In Other words, det A = | A | = = ad-bc.

M2=  = , det (M2) = 16 = 42

M3 =   = , det (M2) = 64 = 43

M4 =    = , det (M2) = 256 = 44

M5 =     = , det (M2) = 1024 = 45

M10 = , det (M2) = 1048576 = 410

M20 = , det (M2) = 1.099511628 x 1012 = 420

M50 = , det (M2) = 1.2676506 x 1030 = 450

By squaring each number in the matrix by the power of Mn you get the answer for each of the matrices. So if M2 = , you then square the matrix, so you multiply it by itself. .        Multiplying the matrix by itself as this: gives you the new matrix which is, in this case .

Middle , det (S2) = 144 = 122 (here k is 12-3)

S3 = = = 22 , det (S3) = 1728 = 123 (here k is (12×2) +3)

S4 = = = 23 , det (S4) = 20736 = 124 (here k is (12×7)-3)

S5 = = = 24 , det (S5) = 248832 = 125 (here k is (12×20) +3)

Given these matrices, we notice that when simplifying the matrices we also use 2n-1 as the factor. We can use the general formula of k to find the numbers inside the new matrix and still a pattern between the matrices’ determinants can be noticed. The determinant of each matrix is 12 powered by the same number you powered the matrix.

If S3 = , this matrix’s determinant is 123, which is 1728, this confirms the trend I stated before. The trend in matrix P is the same in matrix Sbut 4 numbers more. The formula in matrix P is 8n, and 8 + 4 = 12 therefore the formula in matrix S is 12n.

I will make an example for k = 4, so the matrix will be V = .

We can predict according to the determinant’s pattern that the determinants will now be 16n.

V2 = = = 2 , det (V2) = 256 = 162 (here k is 16)

V3 = = = 22 , det (V3) = 4096 = 163 (here k is 16×4)

V4 = = = 23 , det (V4) = 65536 = 164 (herek is 16×16)

V5 = = = 24 , det (V5) = 1048576 = 165(here k is 16×64)

These matrices follow the trend, because 12 + 4 = 16 so the determinants of these matrices must be 16n, where n is the power of the matrix.

Conclusion

k after making a hypothesis about the results of that same value.

For k = 5, the letter should be ‘Y’ according to the letter’s pattern, and according to the formula the matrix should be so you end up with Y = . Now if we power this matrix so as n = 2,3,4,5 we should see the same pattern as the former sets of matrices. The matrices determinants for this set should follow the trend and should be equal to 20n.

Y2 = = = 24 , det (Y2) = 400 = 202

Y3 = = = 25 , det (Y3) = 8000 = 203

Y4 = = = 26 , det (Y4) = 160000 = 204

Y5 = = = 27 , det (Y5) = 3200000= 205

This further value of k, confirms my previous statements about the generalized pattern of these sets of matrices, it also proves the formula and trend of the determinants in each set of matrices.

Conclusions

A general trend was found in the different sets of matrices provided by the assignment sheet. A formula was created, so that it could be applied to generate more matrices that follow the pattern found. The results were satisfactory due to the few limitations that the trend had. It can be considered a limitation that a matrix when powered to 0 gives you always the identity matrix, so n ≠ 0.  Another limitation found is that is useless to power a matrix to 1, because it will remain unchanged. Another fact I found was that when k = negative numbers, the answers are exactly the same as the answers for the positive numbers.

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. ## Math IB HL math portfolio type II. Deduce the formula Sn = ...

= (S2k - Sk) + (S4k - S3k) 2[(3q-3p)-q] = (q - p) + [S4k - (3q - 3p)] 4q - 6p = q - p + S4k - 3q + 3p S4k = 6q - 8p Therefore: S3k = 3q - 3p S4k = 6q - 8p 7.

2. ## Extended Essay- Math

ï¿½kï¿½ï¿½Fï¿½1/4ï¿½ï¿½1UR"Lï¿½jï¿½Hlï¿½_ï¿½9'm&ï¿½fï¿½ï¿½aï¿½(c)1,1/4h5ï¿½,l\Yï¿½ï¿½Ssd+r]ï¿½-ï¿½ï¿½ï¿½ï¿½(tm)Sï¿½ï¿½ncï¿½ï¿½V(c)ï¿½ï¿½ï¿½ï¿½u:ï¿½vï¿½^ï¿½3ï¿½ï¿½(c)ï¿½ï¿½ï¿½ï¿½ï¿½]ï¿½ï¿½ ï¿½ï¿½ï¿½9""][(r)5\|Eï¿½Òï¿½kï¿½Vky-ï¿½ï¿½*5 ï¿½ï¿½1ï¿½_ï¿½ï¿½461/4Yï¿½ï¿½ 3/4ï¿½ï¿½lï¿½W1/4Yï¿½ï¿½ 3/4{ymï¿½ï¿½ï¿½ (f4ÝAï¿½v"ï¿½ï¿½2ï¿½lï¿½ _ï¿½j-ï¿½OQï¿½ evï¿½)ï¿½<Uï¿½r-ï¿½ï¿½ï¿½ï¿½@V4Ej''! !ï¿½ÛÑ z1/4Npï¿½ ï¿½Ìï¿½ ï¿½Pï¿½L5ï¿½2 1ï¿½ï¿½ï¿½w/ï¿½=Aï¿½ï¿½-ï¿½ï¿½ï¿½?XEï¿½ï¿½6NGU(c)ï¿½|ï¿½Ç£ï¿½ï¿½ï¿½O3/4~ï¿½ï¿½s?^Vï¿½|ï¿½ï¿½ï¿½?}ï¿½ Bï¿½Lï¿½yï¿½ï¿½ï¿½_ï¿½>~ï¿½ï¿½'ï¿½}ï¿½ï¿½ï¿½xP...ï¿½iB\$ï¿½I ï¿½-Oï¿½1Êk9ï¿½ï¿½Iï¿½cL"ï¿½ï¿½Hï¿½ï¿½Y<ï¿½;*vï¿½7'ï¿½a(r)E\ï¿½h>ï¿½ï¿½=ï¿½ï¿½^,ï¿½*ï¿½ï¿½ï¿½ï¿½8qï¿½;ï¿½ï¿½^(r)ï¿½*4ï¿½ï¿½ï¿½ï¿½?ï¿½Wq{ï¿½ï¿½nï¿½Ôogï¿½Aß¤>ï¿½ï¿½8fï¿½2ï¿½*<")QHï¿½ï¿½"xï¿½ï¿½K(c)ï¿½ï¿½ -|ï¿½ï¿½]ï¿½Zï¿½z)ï¿½Ó"MSï¿½mï¿½@\&>!ï¿½ 7;wPï¿½3ï¿½ï¿½[ï¿½EBU`ï¿½13/4Oï¿½Cï¿½5<V8ï¿½(c)ï¿½ï¿½"U ï¿½ï¿½Uï¿½3ï¿½7aï¿½' "="ï¿½ï¿½NDï¿½ï¿½ï¿½ï¿½ï¿½o%ï¿½×¡uï¿½ï¿½3/4ï¿½&ï¿½ï¿½ï¿½ï¿½tï¿½ï¿½W'[|ï¿½ï¿½\$ï¿½a{4ï¿½"ï¿½ï¿½>ï¿½(F"\ï¿½ï¿½;Ü­ï¿½ qï¿½ï¿½ï¿½pß¡ï¿½ ï¿½ï¿½ï¿½ï¿½69&MDï¿½ O,ï¿½ï¿½ï¿½ooÂï¿½ï¿½VMï¿½ï¿½ï¿½ M_Õ¸ï¿½Û¹ï¿½×¸ï¿½U>ï¿½ï¿½'ï¿½ï¿½7ï¿½eo 3/4ï¿½ï¿½>Ñ¨ï¿½ï¿½Nï¿½ï¿½6}ï¿½"ï¿½-ï¿½" bvï¿½zÛï¿½6ï¿½ï¿½?ßï¿½ï¿½ï¿½Å·ï¿½iï¿½ï¿½ï¿½Lvï¿½mï¿½ï¿½ï¿½ï¿½]ï¿½ï¿½2ï¿½SFnHï¿½ï¿½-ï¿½ï¿½D]ï¿½rï¿½ï¿½Iï¿½SXï¿½O]ï¿½0ï¿½ï¿½ ldï¿½ï¿½ï¿½Cï¿½ï¿½^ï¿½3Ø´ï¿½ï¿½d\$sï¿½#ï¿½2.ï¿½hï¿½1/2ï¿½5-6ï¿½ï¿½-5-ï¿½!ï¿½v ï¿½ï¿½ ï¿½ï¿½ï¿½'-.ï¿½ï¿½cï¿½ï¿½hï¿½ï¿½-ï¿½ï¿½ï¿½Nï¿½t9W 3/41/2ï¿½dumï¿½(tm)g"ï¿½LStf+]ï¿½ï¿½C9P^ï¿½ï¿½%ï¿½ï¿½(c)Aï¿½-Wï¿½Ì¯ï¿½ï¿½ï¿½ f\$ï¿½1/4ï¿½a1Qï¿½{B"{mï¿½qDlï¿½ï¿½ï¿½ ï¿½u""...fï¿½ï¿½ï¿½ï¿½9ï¿½%[email protected]ï¿½ï¿½Fï¿½ï¿½ï¿½ï¿½?g\$ï¿½P0%OVK"ï¿½ï¿½Rtï¿½ ï¿½--ï¿½ ï¿½ ?" ï¿½&ï¿½6ï¿½ï¿½...JØ¬=ï¿½M'N=^ï¿½gUn.ï¿½ï¿½Sï¿½(tm)ï¿½j ï¿½ï¿½ï¿½ï¿½1/4ï¿½Cï¿½R="ï¿½qï¿½ï¿½"ï¿½bï¿½4"ï¿½Yï¿½ï¿½ )ï¿½Yï¿½vCKï¿½Cï¿½jï¿½+#ï¿½;ï¿½ï¿½wB>VDï¿½ï¿½ï¿½ ï¿½Xï¿½a?pï¿½ï¿½ï¿½ï¿½"ï¿½ Sï¿½(r)ï¿½4ï¿½ï¿½ï¿½[ï¿½NS1/2ï¿½28;ï¿½Yï¿½1/4[ï¿½"(tm)ï¿½ï¿½,ï¿½ï¿½"ï¿½ï¿½T1|ï¿½ï¿½nï¿½;ï¿½+ï¿½ï¿½/Êjï¿½ï¿½\ï¿½ï¿½\-,E:!ï¿½ï¿½ ï¿½tï¿½4.TÌ¡ e1 "ï¿½}ï¿½; [ï¿½z^ï¿½pï¿½lï¿½ï¿½@ï¿½okï¿½ï¿½0e [email protected]ï¿½GGHPXNT,ï¿½...ï¿½dï¿½ï¿½eï¿½|ï¿½*Y(r)ï¿½dTï¿½\(tm)Yï¿½ ä°3/4ï¿½+ï¿½ (ï¿½T7ï¿½\$ow2ï¿½ï¿½ï¿½1/4ï¿½#1/2G(c)Öï¿½ï¿½Ê¥ï¿½ï¿½?1/2qï¿½ï¿½ Nï¿½ï¿½Kï¿½-ï¿½/M,Wï¿½ï¿½ï¿½gï¿½xï¿½FVï¿½/ï¿½"ï¿½FQï¿½â·¶-Oï¿½ï¿½&ï¿½e(r)ï¿½ï¿½ï¿½cï¿½x1/4ï¿½\ Qï¿½ï¿½ï¿½[email protected]ï¿½ï¿½!ï¿½+ï¿½ï¿½ï¿½{ï¿½[|{ï¿½ï¿½!ï¿½ï¿½Kï¿½"Aï¿½i `ï¿½c m2iU-ï¿½|ï¿½Y+ï¿½ Þ¨-ï¿½ [[v-xï¿½"ï¿½rï¿½Nï¿½ï¿½ï¿½E'ï¿½3ï¿½pmï¿½ï¿½R '=Yï¿½04,ï¿½!&0ï¿½+Wï¿½CÜ@oï¿½ï¿½ï¿½ï¿½ï¿½OS2ï¿½'S ï¿½(r)0ï¿½5ï¿½ï¿½\$ï¿½É¤]pmï¿½ï¿½3ï¿½Fï¿½tï¿½ ï¿½ï¿½ï¿½ï¿½Gï¿½"-!ï¿½yï¿½ï¿½"ï¿½ÓV .ï¿½ (r)`ï¿½×¢vï¿½,...O.%ï¿½ï¿½Ð²Kas-ï¿½Sï¿½ï¿½Æ­ï¿½vï¿½ï¿½Mï¿½zï¿½ï¿½"`ï¿½ï¿½...ï¿½3ï¿½ï¿½{ï¿½9+eï¿½ ï¿½ï¿½@1/2eï¿½ï¿½ï¿½"ï¿½Lyï¿½ï¿½

1. ## Math IB SL BMI Portfolio

growth into adult size, and that it is rather a measurement that compares one's height with one weight; it is possible that one could keep gaining weight to a certain point after reaching adult height. This assumption would be that one reaches adult height and adult weight at different times,

2. ## IB math SL type 2 project

The Container of Hot'n'Cold, Inc. Graph 3: The Change of Temperature According to the Equation of y = 50+(210-50)e-0.416x 3. How long does it take each container to lower the coffee temperature from 170 �F to 140 �F? 4. How long will the coffee temperature remain between 120 �F and 140 �F?

1. ## Stellar Numbers. After establishing the general formula for the triangular numbers, stellar (star) shapes ...

Sn = 3n2 - 3n + 1 Un = 3n2 - 3n + 1 U3 = 3(3)2 - 3(3) + 1 U3 = 3(9) - 9 + 1 U3 = 27 - 8 U3 = 19 Un = 3n2 - 3n + 1 U5 = 3(5)2 - 3(5)

2. ## matrix power

Even though we receive a decimal number if the power was odd, the theory is correct and can be applied to other matrices, however the only drawback is the length of the decimal numbers as opposed to a well-rounded whole number.

1. ## MATH IB SL INT ASS1 - Pascal's Triangle

Therefore the general formula for En(r) has to be valid. Nevertheless I tested the validity of the general formula for En(r) one more time. We know that the fractions of the 8th row are and the fractions of the 9th row are: .

2. ## IB Math Methods SL: Internal Assessment on Gold Medal Heights

If there was no war, then this scenario of a reduced value would likely not have existed; it would be likely that the value for 1940 and onwards would keep increasing; and not go down in value until a low in 1948. • Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to 