Question

线性代数...若方阵A与B相似,则|A-|B|E|=|B-|A|E|,为什么?

Answer 1

因为A,B相似,所以存在可逆矩阵P满足 P^-1AP = B
所以 |B| = |P^-1AP| = |P^-1||A||P| = |A|.
且 |B-|A|E|
= |P^-1AP - |B|E|
= |P^-1AP - P^-1(|B|E)P|
= |P^-1(A-|B|E)P|
= |P^-1||A-|B|E||P|
= |A-|B|E|