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# Matrix Binomials Portfolio

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Introduction

Math SL Matrix Binomials Portfolio

This portfolio will investigate the properties of matrix binomials in order to determine a general statement for Mn where n is a real number and an integer, and M is the 2  matrix  Let:

X =  Y=  X2 =  =  X3 =  =  X4 =  Y2 =  Y3 =  Y4 =  (X+Y) =  (X+Y)2 =    (X+Y)3 =  (X+Y)4 =  Expressions for Xn, Yn and (X+Y)n

Xn =  Yn =  (X+Y)n =  n > 0,

let: W = any 2x2 matrix,  W-n =  It is not possible to

Middle   B3 =  B4 =  Therefore:

(A+B) =  (A+B)2 =      (A+B)3 =    (A+B)4 =    Expressions for An, Bn and (A+B)n

An =  Bn =  (A+B)n =  n > 0,

let: W = any 2  2 matrix,  W-n =  It is not possible to divide an integer by a matrix, n < 0 does not exist

n≠0

For any matrix where n=0 Wn = I    W0 =  Let: M =  , M = A+B and M2 = A2+B2

A = aX =  Conclusion

n must also be greater than zero. It cannot equal zero because any matrix raised to the power of zero would equal I the identity matrix, and it cannot be less than zero, because that would be the same as one divided by the matrix raised to the power of n. Since it is not possible to divide an integer by a matrix, it is not possible to raise any matrix to a negative exponent and this n must be greater than zero. Therefore the expressions determined in this portfolio are valid for 2  matrices, where the value of n is greater than zero

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

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