[線代] 一題證明

[tim8238818 (AAAAAAAAAAAAAAAAAAAAAAA)]

Let A, B and C be n x n matrices. We say that A is similar to B if there is
an n x n non-singular matrix P, such that（P-1）AP = B. Prove each of the
following statements. （P-1是P的inverse）

a. If A is similar to B, then B is similar to A.
b. If A is similar to B and B is similar to C, then A is similar to C

第一小題我寫
A=B
A=(p-1)AP
PA=AP substitute to (P-1)AP=B
(P-1)(PA)=B
A=B


第二小題

A=B and B=C=(P-1)AP
A=(P-1)AP
PA=AP substitute to (P-1)AP=C
thus (P-1)PA=C
A=C


不知道這樣對不對，先謝謝板上神人賜教

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→ egg12388    :錯，為什麼一開始就有"A=B"??
→ egg12388    :順帶一提，P的反矩陣可以打成 "P^{-1}"    "^"是次方
→ egg12388    :定義"similar"有它的意義，並不能保證得到 "equal"
→ tim8238818  :謝謝樓上指正所以我的起手式要怎麼寫啊@@
→ jimmy86204  :因為A和B相似.........
→ egg12388    :所以P^{-1}AP = B for some invertible matrix P.
→ egg12388    :你只要想辦法找到另一個可逆矩陣Q滿足Q^{-1}BQ=A就好