# Variance covariance matrix for the inverse of a matrix

Suppose we have a matrix $\mathbf{A}=\begin{bmatrix}a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix}$ and know its variance covariance $\left(4\times4\right)$ matrix. Then how the variance covariance matrix of $\mathbf{A}^{-1}$ can be obtained? I would highly appreciate your help. Thanks

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What do you call a variance covariance matrix of a matrix ? –  Stéphane Laurent Jul 21 '12 at 7:53
@StéphaneLaurent I suppose that he meant the covariance of a DATA matrix. In this case, if we label $\vec{a}_1=(a_{11},a_{12})^T$ and $\vec{a}_2=(a_{21},a_{22})^T$, then $\mathbf{A}=(\vec{a}_1,\vec{a}_2)^T$ and the sample variance-covariance matrix the OP's refereing to would have elements $\mathbf{\Sigma}_{ij}=\text{Cov}(\vec{a}_{i},\vec{a_{j}})$. –  Néstor Jul 21 '12 at 8:13
How can a (data?) matrix 2x2 have covariance matrix 4x4? –  ttnphns Jul 21 '12 at 13:22
@Néstor has a good explanation. Thanks –  MYaseen208 Jul 21 '12 at 15:11
@MYaseen208, ttnphns is right. If it is the covariance of a data matrix, it should be 2x2. –  Néstor Jul 21 '12 at 18:50