Help Proving Linear Regression Model in Matrix Notation I need to show that: 
$\hat \mu ^{'} \hat \mu = \hat \mu ^{'} y$ 
Given the matrix notation of the linear regression model: $y= X\beta + \mu$
Also given: $\hat y = X \hat \beta $ and $\hat \mu = y - X\hat \beta $
I have tried:
$(y-X \hat \beta)^{'} (y-X\hat \beta)$ = $ (y-X \hat \beta)^{'} y $
I'm not sure if there's a property of transpose that I'm not aware of but I don't see how we could be saying $(y-X \hat \beta) = y $ 
Are we assuming that $X\hat\beta = 0 $ ?

We know that $$X'(y-X\hat{\beta})=0\tag{1}$$
We want to show that $$\hat{\mu}'\hat{\mu}=\hat{\mu}'y$$
which is equivalent to 
$$(y-X\hat{\beta})'X\hat{\beta}=0$$
Let's take the transpose,
$$\hat{\beta}' \color{red}[X'(y-X \hat{\beta})\color{red}]=0$$
From $(1)$, we can see that the result is true.
Regarding your attempt:
For matrices, if $AB=AC$, we can't conclude that $B=C$ unless there are other conditions such as $A$ is a non-singular matrix.
