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
$d A_{11} + eA_{12} + fA_{13}=(-1)^{1 +1} d(ei - fh) + (-1)^{1 +2} e(di - fg) + (-1)^{1 +3}f(dh - eg)=dei -dfh -dei +efg + fdh - efg=0$
