Permutation Matrix

what happens to the inverse of covariance matrix if I permute the random vector y?

Suppose var(y)=R, and it is pre-multiplied by a permutaion matrix $K$, then

$$var^{-1}(Ky)=KR^{-1}K^{T}$$

because

1.

$$var(Ky)=KRK^{T}$$

2.

$$K^TK=I\\ KK^T=I$$

3.

$$\begin{align} (KRK^{T})(KR^{-1}K^{T}) &=KR(K^{T}K)R^{-1}K^T\\ &=KRR^{-1}K^T\\ &=KK^T\\ &=I \end{align}$$