Element by element formulae for 3x3 matrix inversion

Question

Given a 3 x 3 matrix:

$$ A= \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix} $$

Can $A^{-1}$ be shown as as a 3x3 matrix with each element in terms of $a,b,c,d,e,f,g,h$ and $i$. Showing basic operators (only $+ - / *$)?

Answer 1

$$A^{-1}=\frac{1}{\det(A)} adj(A)$$

$$\det(A)=aei+dhc+bfg-ceg-bdi-afh$$

$$adj(A)= \begin{bmatrix} ei-fh & ch-bi & bf-ce \\ fg-di & ai-cg & dc-af \\ dh-ge & bg-ah & ae-bd \\ \end{bmatrix}$$

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