A bad Cayley–Hamilton theorem proof [duplicate]

Given $A\in M_{n \times n}(\mathbb{F})$ and $p_{A}(x)=\det(xI-A)$ why saying that $\det(AI-A)=0$ is not valid?

It's not valid because in the expression $xI$, the hidden operation is scalar multiplication, i.e., multiplication of the identity matrix by the constant real value $x$. In the expression $AI$, the hidden operation is matrix multiply, which is in general different from scalar multiply.

There's a real subtlety here: the polynomial $p_A(x)$ is, as defined by the equation, a polynomial function of a single real variable. The CH theorem says that if you, after having computed the coefficients of that polynomial, now plug in the matrix $A$ for the variable $x$, and treat powers of $x$ as matrix powers of $A$, you get the zero matrix. Doing that plugging=in=a=matrix=where=a=real=number=should=be is a very weird operation, and so the result is quite surprising.