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_{i j} \end{array}

be the (n – 1) x (n – 1) matrix obtained by deleting the

\begin{array}{l} i^{t h} \end{array}

row and

\begin{array}{l} j^{t h} \end{array}

column. Then,

\begin{array}{l} d e t M_{i j} \end{array}

is called the minor of

\begin{array}{l} a_{i j} \end{array}

. The cofactor

\begin{array}{l} A_{i j} \end{array}

\begin{array}{l} a_{i j} \end{array}

is defined by:

$A_{i j} = (- 1)^{i + j} det M_{i j}$

Solved Examples

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

Then,

\begin{array}{l} M_{32} = (\begin{array}{c} 2 & - 1 \\ 0 & 4 \end{array}) \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}

FORMULAS Related Links
Important Math Formulas	A2 B2 C2 Formula
Riemann Sum Formula	Charles Law Formula
Diagonal Of A Cube	Mass Defect Formula
Convert Moles To Grams	Ratio And Proportion Formula
Perpendicular Formula	Henry's Law Formula

FORMULAS Related Links

Important Math Formulas

A2 B2 C2 Formula

Riemann Sum Formula

Charles Law Formula

Diagonal Of A Cube

Mass Defect Formula

Convert Moles To Grams

Ratio And Proportion Formula

Perpendicular Formula

Henry's Law Formula

Cofactor Formula

Cofactor Formula

Solved Examples

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