In a solution from a physics problem, the author uses (for $M$ a 3x3 matrix) that $$M_{33} = \frac{|M_\perp^{-1}|}{|M^{-1}|}$$ where $M_\perp^{-1}$ is the upper 2x2 block of $M^{-1}$. This bears a vague resemblance to Cramer's rule but I'm not sure? It also seems like a Laplace expansion rearranged, but other terms are missing? The presence of the inverses is confusing.
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The determinental formula for the inverse of a matrix (closely related to Cramer's rule) is
$$ A^{-1}_{ji} = (-1)^{i+j} \frac{\det(A \text{ with the $i$th row + $j$th column deleted)}}{\det A}. $$
*Notice that's $A^{-1}_{ji}$ not $A^{-1}_{ij}$.
For $i = j = 3$ you get exactly your formula but with $A$ and $A^{-1}$ trading places which is justified because $A$ is the inverse of $A^{-1}$.
Trevor Gunn
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Superb, thank you so much! – EE18 Aug 04 '23 at 02:41