Cofactor matrix

The cofactor matrix (denoted by cof) is the matrix created from the determinants of the matrices not part of a given element's row and column. For example,

${\text{cof}}{\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}}={\begin{bmatrix}{\begin{vmatrix}e&f\\h&i\end{vmatrix}}&-{\begin{vmatrix}d&f\\g&i\end{vmatrix}}&{\begin{vmatrix}d&e\\g&h\end{vmatrix}}\\-{\begin{vmatrix}b&c\\h&i\end{vmatrix}}&{\begin{vmatrix}a&c\\g&i\end{vmatrix}}&-{\begin{vmatrix}a&b\\g&h\end{vmatrix}}\\{\begin{vmatrix}b&c\\e&f\end{vmatrix}}&-{\begin{vmatrix}a&c\\d&f\end{vmatrix}}&{\begin{vmatrix}a&b\\d&e\end{vmatrix}}\end{bmatrix}}$

Notice that the elements of the matrix follow a "checkerboard" pattern of positives and negatives.

A related type of matrix is an adjoint or adjugate matrix, which is the transpose of the cofactor matrix. Adjoints are very useful in computing inverses of matrices.