Show that every matrix X in M(nn) has the form X=A^t-2A for some matrix A in M(nn). Hint: Use the Dimension Theorem.

Answer 1

Let f(A)=A^t-2A. This is a linear map from M(n) to M(n). Ker(f)=0 because if f(A)=0 then A^t-2A=0. So, A^t=2A and this is only possible if A=0 (prove it!). Therefore, by the dimension theorem dim(M(n))=dim(Ker(f))+dim(Im(f))=0+dim(Im(f))=dim(Im(f)). In other words M(n) is isomorphic to Im(f). That is for every matrix X there is a matrix A such that X=f(A)=A^t -2A as requested.