# Portfolio SL - Matrix

MATRIX BINOMIALS

In this portfolio I will try to find a general statement and patterns for given matrix binomials exercises. For data processing I will use TI-83 Plus (Sliver Edition) graphing calculator. I will use my knowledge from patterns and matrix  in order to find suitable formulas.

X=, Y=.

Find X2, X3, X4; Y2, Y3, Y4.

X2==              X3=        X4=
Y2=                Y3=                               Y4=

Find expressions for Xn , Yn, (X+Y)n

• The entries double for every higher power of X, i.e.:

X2= 2X= 21X
X
3= 4X= 22X
X
4= 8X = 23X     follows
Xn=2n-1X

I will test this formula with a random number:                                    X10=               X10= 210-1=

X19= 219-1=

Xn=2n-1X formula valid for all natural numbers N;  = {1,2,3,...} as it is shown in examples.

• The entries double for every higher power of Y, i.e.:

Y2= 2Y
Y
3= 4Y= 22Y
Y
4= 8Y = 23Y     follows
Yn=2n-1Y

Again I will test is with random numbers using GDC:

Y8=              Y8=28-1=

Y16=    Y16=216-1

Y
n=2n-1Y formula valid for all natural numbers N;  = {1,2,3,...} as it is shown in examples.

• In the case of (X+Y)= 2I
(X+Y)
= = =2I
(X+Y)
n= n =  n = = 2n

In this case we have two slightly different formulas, the first one includes the sum of (X+Y) and the second one includes the identify matrix, i.e. I just factorized 2n.

= 2n

Testing with a random number and checking it with GDC.

(X+Y)10= =
(X+Y)10= 210=

(X+Y)15= 215=
(X+Y)15= =
2nformula valid for all natural numbers N;  = {1,2,3,...} as it is shown ...