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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.

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