1.Show that trace of AB=trace of BA

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

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

> A.B;

> trace(A.B);

> B.A;

> trace(B.A);

¬Gtrace(A.B)=trace(B.A)

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

> (A.B-B.A)^2;

> 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));

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

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

¬G(AB-BA)^2¬Oscalar matrix

3.Find all matrices commuting with

> A.C-C.A;

¬Gb=0,c=0

>

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);

5.Show that there is no real invertible matrix S with

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

> S:=Matrix([[0,a],[b,c]]);

> S^(-1).F.S;

>

6. 3x+1 Problem

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

7.The number pattern to the derivatives of tan(x)

8.The number pattern to the derivatives of cot(x)

>

9. Express tan(nx)

> expand(tan(x+y));

> for n from 1 to 6 do expand(tan(n*x)) od;

>

10. Study the number pattern associated with the indefinite integral of the function x^n.e^ x

> for n to 6 do factor(int(x^n*exp(x),x)) od;

>

11. Chebyshev polynomials Tn(x)

> for n to 6 do expand(cos(n*x)) od ;

12. Find an exponent n such that some coefficients appearing in the factorization of (x^n)-1 are different from 0,(-1),1

> factor(x^2001-1);

13. factor((((x^2-a)^2-a)^2-a)^2-a-x)

> factor((((x^2-a)^2-a)^2-a)^2-a-x);