Experiments with Two-by-Two Matrices

1.show that trace of AB= trace of BA

> A:=Matrix([[a,b],[c,d]]);

> B:=Matrix([[p,q],[r,s]]);

> A.B;

>

> B.A;

> A.B-B.A;

>

2.show that (AB-BA)(AB-BA) is scalar

>

> (A.B-B.A).(A.B-B.A);

>

> expand(((b*r-c*q)^2+(a*q+b*s-p*b-q*d)*(c*p+d*r-r*a-s*c))-((a*q+b*s-p*b-q*d)*(c*p+d*r-r*a-s*c)+(c*q-b*r)^2));

>

3.find all matrices commuting with [[1,0][0,2]]

>

> C:=Matrix([[1,0],[0,2]]);

> A.C-C.A;

>

4.show that A^(-1)BA can never be diagonal

>

>

> A^(-1).B.A;

> expand((d*p/(a*d-b*c)-b*r/(a*d-b*c))*b+(d*q/(a*d-b*c)-b*s/(a*d-b*c))*d);

> expand((-c*p/(a*d-b*c)+a*r/(a*d-b*c))*a+(-c*q/(a*d-b*c)+a*s/(a*d-b*c))*c);

>

show that there is no real invertible matrix A with A^(-1)[[1,0],[1,1]]A triangular

>

>

> E:=Matrix([[1,0],[1,1]]);

> A^(-1).E.A;

> expand((d/(a*d-b*c)-b/(a*d-b*c))*b-b*d/(a*d-b*c));

>

> expand((-c/(a*d-b*c)+a/(a*d-b*c))*a+a*c/(a*d-b*c));

Spreadsheet in Maple

1.use the spreadsheet in Maple to experiment with the 3x+1 problem

> restart;

> f:=x-> if x mod 2=0 then x/2 else 3*x+1 fi;

> f(3);

> f(10);

>

>

1.How is the number pattern related to the

> restart;

3.How is the number pattern related to the sucessive derivatives of cot(x) formed?

>

4.Express tan(nx) as a rational function of tan(x) for n=1,2,3,4,5,6

>

5.Study the number pattern associated with the indefinite integral of the function x^ne^x

>

>

6.list the Chebyshev polynomials Tn(x) for n = 0,1,2,3,4,5,6

>