Covariance times constant, basic rules

Question

I know the from the basic rule of the covariance we have: $$\text{Cov(aX,Y)=aCov(X,Y)}$$ however now i'm looking at a case that is creating me some doubt: Looking at the covariance of the same random variable:

$1)$ $\text{Cov(aX,X)=aCov(X,X)=aVar(X)}$

$2)$ $\text{Cov(aX,X)=Var(aX)=}a^2\text{Var(X)}$

which one is the correct solution?

Thank you in advance

Answer 1

The first is correct.

The second is not: By definition $\operatorname{Cov}(X,X)=\operatorname{Var}(X)$.

So $$\operatorname{Var}(aX)= \operatorname{Cov}(aX,aX)= a\operatorname{Cov}(X,aX)= a^2\operatorname{Cov}(X,X)=a^2 \operatorname{Var}(X)$$

So the last $=$ of 2 is correct, the first is not.