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I am working with a Linear Code, C, over $F_{2^5}$, and its dual, $C^{\perp}$. I have the generator matrix for $C$, $G$, and have calculated the generator matrix for $C^{\perp}$, $G^{\perp}$. I need to find a permutation matrix $P$ such that $GP = G^{\perp}$. I have determined that, if $G = (I_2\ |\ A)$, then $G^{\perp} = (I_2\ |\ A^{-1})$. However, I have no idea how to invert only part of a matrix, and was hoping someone could get me started in the right direction.

This is a homework problem, so I am not asking for a solution or anything, I have just run out of ideas and was hoping for some new ones.

Thanks in advance.

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