Cofactor Formula
A Cofactor, in mathematics, is used to find the inverse of the matrix, adjoined. The Cofactor is the number you get when you remove the column and row of a designated element in a matrix, which is just a numerical grid in the form of rectangle or a square.
The cofactor is always preceded by a positive (+) or negative (-) sign. Let A be an n x n matrix and let \(\begin{array}{l}M_{ij}\end{array} \)Â be the (n – 1) x (n – 1) matrix obtained by deleting the \(\begin{array}{l}i^{th}\end{array} \)Â row and \(\begin{array}{l}j^{th}\end{array} \)Â column. Then, \(\begin{array}{l}detM_{ij}\end{array} \)Â is called the minor of \(\begin{array}{l}a_{ij}\end{array} \). Â The cofactor \(\begin{array}{l}A_{ij}\end{array} \)Â of \(\begin{array}{l}a_{ij}\end{array} \)Â is defined by:
\[\LARGE A_{ij}=(-1)^{i+j}\; \det\;M_{ij}\]
Solved Examples
Example: LetÂ
\(\begin{array}{l}\large \begin{pmatrix} 2 & 5 & -1\\ 0 & 3 & 4\\ 1 & -2 & -5 \end{pmatrix}\end{array} \)
 Then,
\(\begin{array}{l}\large M_{32}=\bigl(\begin{matrix} 2 & -1\\ 0 & 4 \end{matrix}\bigr)\end{array} \)
So the minor of
\(\begin{array}{l}a_{32}\end{array} \)
 is the determinant of this 2 x 2 matrix.Since the matrix is triangular, the determinant is the product of the diagonals.
(2) (4) = 8
\(\begin{array}{l}A_{32}=(-1)^{3+2}(8)=-8\end{array} \)
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