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

\(\begin{array}{l}M_{ij}\end{array} \)
 be the (n – 1) x (n – 1) matrix obtained by deleting the
\(\begin{array}{l}i^{th}\end{array} \)
 row and
\(\begin{array}{l}j^{th}\end{array} \)
 column. Then,
\(\begin{array}{l}detM_{ij}\end{array} \)
 is called the minor of
\(\begin{array}{l}a_{ij}\end{array} \)
.  The cofactor
\(\begin{array}{l}A_{ij}\end{array} \)
 of
\(\begin{array}{l}a_{ij}\end{array} \)
 is defined by:

\[\LARGE A_{ij}=(-1)^{i+j}\; \det\;M_{ij}\]

Solved Examples

Example: Let 
\(\begin{array}{l}\large \begin{pmatrix} 2 & 5 & -1\\ 0 & 3 & 4\\ 1 & -2 & -5 \end{pmatrix}\end{array} \)
 
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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