Deriving with matrices, matrice equation

Question

Can someone do this stepwise so that I am able to see what is going on. I am sure there is not many steps to take, but I am struggling to see the logic that goes from the left hand side to the right.! $$(X^T X)^{-1}X^T(\sigma^2I_n)X(X^TX)^{-1} = \sigma^2(X^TX)^{-1}$$

Answer 1

So, as has already been said in the comments, you can pull the scalar $\sigma^2$ all the way to the left. In addition to this, we can leave off $I_n$ because it is the identity (i.e. it does nothing by definition). This gives us:

$$\begin{eqnarray} (X^T X)^{-1}X^T(\sigma^2I_n)X(X^TX)^{-1} =& \sigma^2(X^T X)^{-1}X^TX(X^TX)^{-1} \\ =& \sigma^2(X^T X)^{-1}\underbrace{(X^TX)(X^TX)^{-1}}_{=I_n} \\ =& \sigma^2(X^T X)^{-1} \end{eqnarray}$$

Which is what was to be shown.