Is there any closed-form expression to calculate each element of the inverse of a matrix?

Considering a generic square matrix $A=(a_{i,j})$ we want to compute its inverse $A^{-1}=\left[a^{(-1)}_{i,j}\right]$.

Is there a way to express each $a^{(-1)}_{i,j}$ using a closed form expression?

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– Rahul Narain Jun 14 '12 at 2:23

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Depends on what you mean by closed form. Cramer's rule involves det. – André Nicolas Jun 14 '12 at 2:29
Yeah by closed-form expression I mean a set of rules that involves elementary operations... For example, cofactors are calculated using minors, If I wanted to replace the cofactor term in the relation with an expression, how would it be... What I'd like to reach is a final formula not involving more steps to calculate the final quantity. Maybe it is not possible, just want a confirmation of this if possible. – Andry Jun 14 '12 at 2:46

1 Answer

The $ij$ entry of $A^{-1}$ is $(-1)^{i+j}$ times the determinant of the matrix $C_{ji}$ obtained by deleting row $j$ and column $i$ from $A$, all divided by the determinant of $A$. I don't know whether you consider that to be a closed form.

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Since the formula is a rational function in the entries of $A$, it should be good :) – N. S. Jun 14 '12 at 2:48
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