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

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

> B := _rtable[18397620];

> B := _rtable[18397620];

> B;

Find all matrices commuting with

> A.B;

> B.A;

> A.B-B.A;

Ans: All diagonal matrix.

Show that trace of AB = trace of BA

with(linalg);

Warning, the protected names norm and trace have been redefined and unprotected

> trace(A.B);

> trace(B.A);

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));

Show that A-1BA 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);

Ans: When B = c I , A^(-1)*B*A can be diagonal.

Show that there is no real invertible matrix S with S-1 S triangular.

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

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

Ans: If S= , then S(-1)* *S is triangular .