Let with . Then there exists unique matrix such that AB=BA=det(A)I and B has integer entries.
Help me to prove this.
the matrix is the adjugate of the proof of can be found in any linear algebra textbook. the only thing that i have to add here is that if
then from the definition of the adjugate matrix, it's clear that as well. so i don't know why your problem says ? we also don't need to have
Edit: just realized that we want a unique in this case, the condition will take care of that.