Editing Adjugate matrix (section)

=== 3 × 3 numeric matrix ===
As a specific example, we have
:<math>\operatorname{adj}\!\begin{bmatrix}
-3 &  2 & -5 \\
-1 &  0 & -2 \\
 3 & -4 &  1
\end{bmatrix} = \begin{bmatrix}
-8 & 18 & -4 \\
-5 & 12 & -1 \\
 4 & -6 &  2
\end{bmatrix}.</math>
It is easy to check the adjugate is the [[inverse matrix|inverse]] times the determinant, {{math|−6}}.

The {{math|−1}} in the second row, third column of the adjugate was computed as follows.  The (2,3) entry of the adjugate is the (3,2) cofactor of '''A'''.  This cofactor is computed using the [[submatrix]] obtained by deleting the third row and second column of the original matrix '''A''',
:<math>\begin{bmatrix} -3 & -5 \\ -1 & -2 \end{bmatrix}.</math>
The (3,2) cofactor is a sign times the determinant of this submatrix:
:<math>(-1)^{3+2}\operatorname{det}\!\begin{bmatrix}-3&-5\\-1&-2\end{bmatrix} = -(-3 \cdot -2 - -5 \cdot -1) = -1,</math>
and this is the (2,3) entry of the adjugate.