Last updated: Jan. 2nd, 2019
Let $A$ be a 3x3 matrix given by

The inverse matrix is

, where
$|A| \neq 0$.

Proof

There are mainly two ways to obtain the inverse matrix.
One is to use Gauss-Jordan elimination and the other is to use the adjugate matrix.
We employ the latter, here.

The inverse matrix has the property that
it is equal to the product of the reciprocal of the determinant and the adjugate matrix.

, where $\tilde{A}$ is the adjugate matrix of $A$, and
$|A|$ is the determinant

. (See determinant of a 3x3 matrix.)
The adjugate matrix $\tilde{A}$ is obtained as follows.
Let $M_{ij}$ be a submatrix obtained by removing $i$-th row and $j$-th column from $A$,

These determinants are

Each element of the adjugate matrix $\tilde{A}$ is defined as

(Note that the subscript order of $M$ is different from that of $\tilde{A}$).
Therefore, we have

and in the form of a matrix,

By substituing $|A|$ and $\tilde{A}$ into $(1)$, we obtain

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