 # Minors and Cofactors of Determinants

## Minors of Determinants

For a given square matrix $A$ of order $n \times n$, the minor $M_{ij}$ corresponding to the element $a_{ij}$ is the determinant of the submatrix that remains after removing the $i^{th}$ row and $j^{th}$ column from $A$.

For a 2x2 matrix
$A = \begin{vmatrix} a & b \\ c & d \end{vmatrix}$

The minors are simply the elements themselves, as removing a row and a column leaves a 1x1 matrix.
$M_{11} = a$
$M_{12} = b$
$M_{21} = c$
$M_{22} = d$

For a 3x3 Matrix
Consider a 3x3 matrix $A$ given by
$A = \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}$
The minor $M_{11}$ corresponding to $a$ is the determinant of the 2x2 matrix obtained by removing the first row and first column:
$M_{11} = \begin{vmatrix} e & f \\ h & i \end{vmatrix} = ei - fh$

The minors $M_{ij}$ can be calculated as follows:
- $M_{11} = ei - fh$
- $M_{12} = di - fg$
- $M_{13} = dh - eg$

## Cofactors of Determinants

Cofactor of an element $a_{ij}$ , denoted by $A_{ij}$ is defined by
$A_{ij} = (–1)^{i + j} M_{ij}$ , where $M_{ij}$ is minor of $a_{ij}$

For a 2x2 matrix
$A = \begin{vmatrix} a & b \\ c & d \end{vmatrix}$
Minors are given as
$M_{11} = d$
$M_{12} = c$
$M_{21} = b$
$M_{22} = a$
Hence Cofactors are
$A_{11} = d$
$A_{12} = -c$
$A_{21} = -b$
$A_{22} = a$

For a 3x3 Matrix
Consider a 3x3 matrix $A$ given by
$A = \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}$
The minors $M_{ij}$ can be calculated as follows:
- $M_{11} = ei - fh$
- $M_{12} = di - fg$
- $M_{13} = dh - eg$
Hence cofactors will be
- $A_{11} = ei - fh$
- $A_{12} = -(di - fg)$
- $A_{13} = dh - eg$

## Determinants Value in terms of Cofactors

The determinant $|A|$ is calculated as
$|A| = (-1)^{1 +1} a(ei - fh) + (-1)^{1 +2} b(di - fg) + (-1)^{1 +3}c(dh - eg)=a A_{11} + bA_{12} + cA_{13}$
Similary we can can be calculated by other five ways of expansion that is along R2, R3,C1, C2 and C3
Hence $\Delta$ = sum of the product of elements of any row (or column) with their corresponding cofactors
Important Point
If elements of a row (or column) are multiplied with cofactors of any other row (or column), then their sum is zero. For example
$d A_{11} + eA_{12} + fA_{13}=(-1)^{1 +1} d(ei - fh) + (-1)^{1 +2} e(di - fg) + (-1)^{1 +3}f(dh - eg)=dei -dfh -dei +efg + fdh - efg=0$

• Notes NCERT Solutions & Assignments

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