About a very elementary method which determines if $A$ is similar to $B$ or not. Please tell me a linear algebra book which includes this method.

I am reading "LINEAR ALGEBRA WITH Mathematica" (in Japanese) by Yoshiharu Taniguchi and Kiyokazu Nagatomo.

The following elementary method which determines if $A$ is similar to $B$ is in this book.

Let $A$ and $B$ be an $n\times n$ matrix.
If $A$ is similar to $B$, then there exists a non-singular $n\times n$ matrix $P=(p_{ij})$ such that $$P^{-1}AP=B.$$ If $A$ is similar to $B$, then there exists a non-singular $n\times n$ matrix $P=(p_{ij})$ such that $$AP-PB=O.$$
$AP-PB=O$ is $n^2$ linear equations in the $n^2$ variables $p_{11},p_{12},\dots,p_{nn}$.
So, we can solve this system of linear equations easily.
Let $P_1,\dots,P_r$ be a basis of the space of the solutions of this system of linear equations.
If there are scalars $c_1,\dots,c_r$ such that $$\det(c_1P_1+\dots+c_rP_r)\neq 0,$$ then $A$ is similar to $B$.
If $$\det(c_1P_1+\dots+c_rP_r)=0,$$ for all scalars $c_1,\dots,c_r$, then $A$ is not similar to $B$.

I think many authors don't write this elementary method in their linear algebra books.
I wonder why they don't write this elementary method.
They don't write this elementary method because this method is very obvious?
Is there any linear algebra book which includes this elementary method?

Let $\psi$ denote a linear transformation $\psi(X)=AX-XB,V\to V$. We know that the representation matrix of the mapping is $A\bigotimes I-I\bigotimes B $ (use the method of elementary matrices), so it is reasonable to introduce Kronecker Product first.

Also, since Jordan canonical form is more convenient and commonly used , i think it's not necessary to discuss this method.