(12) Let us try to understand the equation AB-BA=M. Let F be any field and let A and
B be two matrices in M_(n imes n)(F). Assume that F is an algebraically closed field or you
can solve the problem for any field by using the fact that every field F is a subfield of an
algebraically closed field.
(a) Assume that F=C. Show that if A and B are both diagonalisable then M=A has
no non-zero solutions.
(b) Assume that M is a matrix with M=(m_(ij)) such that Tr(M)=0 and sum_(i=1)^(n-1) m_(i,i+1)=
0 and m_(ij)=0 for all j>=i+2. Show that AB-BA=M has a solution. (Hint:
Consider the matrices K=(k_(ij)) and B=(b_(ij)) such that k_(j+1,j)=1 for all k_(ij)=0;b_(i1)=0b_(i,i+3)=0MAB-BA=M1<=j
and all other k_(ij)=0;b_(i1)=0 and b_(i,i+3)=0
12) Let us try to understand the equation AB - BA = M. Let F be anv field and let A and B be two matrices in Mnxn(F). Assume that F is an algebraically closed field or you can solve the problem for any field by using the fact that every field F is a subfield of an algebraically closed field. a) Assume that F = C. Show that if A and B are both diagonalisable then M = A has no non-zero solutions.
0 and mii = 0 for all j i + 2. Show that AB - BA = M has a solution.(Hint: Consider the matrices K = (kij) and B = (bij) such that kj+1,j = 1 for all 1 j < n and all other kij =0; bi1 =0 and bi,i+3 =0) (c) Show that any matrix M with trace zero has solutions to the equation AB-BA = M