The
cofactor of the element \(a_{ij}\) is denoted by \(A_{ij}\) and is defined by
\[\boxed{A_{ij}=(-1)^{i+j}M_{ij}}\]
Thus, the cofactor is obtained by multiplying the corresponding minor by the appropriate sign factor.
Cofactor Sign Pattern
The signs of the cofactors in a \(3\times3\) determinant follow the checkerboard pattern:
\[\begin{bmatrix}+ & - & +\\- & + & -\\+ & - & +\end{bmatrix}\]
This sign pattern is extremely important when expanding determinants.
Memory rule: Start with \(+\) in the top-left corner and alternate signs horizontally and vertically.
Expansion of a Determinant Along the First Row
Consider
\[|A|=\begin{vmatrix}a_{11} & a_{12} & a_{13}\\a_{21} & a_{22} & a_{23}\\a_{31} & a_{32} & a_{33}\end{vmatrix}\]
When expanding along the first row, each element of \(R_1\) is multiplied by its corresponding cofactor.
Therefore,
\[|A|=a_{11}A_{11}+a_{12}A_{12}+a_{13}A_{13}\]
Since
\[A_{11}=M_{11},\]
\[A_{12}=-M_{12},\]
and
\[A_{13}=M_{13},\]
we obtain
\[\begin{aligned}|A|&=a_{11}\begin{vmatrix}a_{22} & a_{23}\\a_{32} & a_{33}\end{vmatrix}-
a_{12}\begin{vmatrix}a_{21} & a_{23}\\a_{31} & a_{33}\end{vmatrix}\\&\quad+
a_{13}\begin{vmatrix}a_{21} & a_{22}\\a_{31} & a_{32}\end{vmatrix}\end{aligned}\]
Fully Expanded Formula
Using the \(2\times2\) determinant formula for each minor,
\[\begin{aligned}|A|&=a_{11}(a_{22}a_{33}-a_{23}a_{32})\\&\quad-a_{12}(a_{21}a_{33}-a_{23}a_{31})\\&\quad+a_{13}(a_{21}a_{32}-a_{22}a_{31})\end{aligned}\]
Hence, the standard expansion along the first row is
\[\begin{aligned}|A|&=a_{11}(a_{22}a_{33}-a_{23}a_{32})-a_{12}(a_{21}a_{33}-a_{23}a_{31})\\&\quad+a_{13}(a_{21}a_{32}-a_{22}a_{31})\end{aligned}\]
Expansion Along the Second Row
For
\[|A|=\begin{vmatrix}a_{11} & a_{12} & a_{13}\\a_{21} & a_{22} & a_{23}\\a_{31} & a_{32} & a_{33}\end{vmatrix},\]
expansion along \(R_2\) gives
\[|A|=a_{21}A_{21}+a_{22}A_{22}+a_{23}A_{23}\]
The signs along the second row are
\[-,\quad +,\quad -\]
Hence,
\[\begin{aligned}|A|&=-a_{21}\begin{vmatrix}a_{12} & a_{13}\\a_{32} & a_{33}\end{vmatrix}+
a_{22}\begin{vmatrix}a_{11} & a_{13}\\a_{31} & a_{33}\end{vmatrix}\\&\quad-
a_{23}\begin{vmatrix}a_{11} & a_{12}\\a_{31} & a_{32}\end{vmatrix}\end{aligned}\]
Using \(2\times2\) determinant evaluation,
\[\begin{aligned}|A|&=-a_{21}(a_{12}a_{33}-a_{13}a_{32})\\&\quad+a_{22}(a_{11}a_{33}-a_{13}a_{31})\\&\quad-a_{23}(a_{11}a_{32}-a_{12}a_{31})\end{aligned}\]
Expansion Along the Third Row
Expansion along \(R_3\) gives
\[|A|=a_{31}A_{31}+a_{32}A_{32}+a_{33}A_{33}\]
The signs in the third row are
\[+,\quad -,\quad +\]
Therefore,
\[\begin{aligned}|A|&=a_{31}\begin{vmatrix}a_{12} & a_{13}\\a_{22} & a_{23}\end{vmatrix}-
a_{32}\begin{vmatrix}a_{11} & a_{13}\\a_{21} & a_{23}\end{vmatrix}\\&\quad+
a_{33}\begin{vmatrix}a_{11} & a_{12}\\a_{21} & a_{22}\end{vmatrix}\end{aligned}\]
Expansion Along the First Column
Expansion along \(C_1\) gives
\[|A|=a_{11}A_{11}+a_{21}A_{21}+a_{31}A_{31}\]
The signs along the first column are
\[+,\quad -,\quad +\]
Hence,
\[\begin{aligned}|A|&=a_{11}\begin{vmatrix}a_{22} & a_{23}\\a_{32} & a_{33}\end{vmatrix}-
a_{21}\begin{vmatrix}a_{12} & a_{13}\\a_{32} & a_{33}\end{vmatrix}\\&\quad+
a_{31}\begin{vmatrix}a_{12} & a_{13}\\a_{22} & a_{23}\end{vmatrix}\end{aligned}\]
Expansion Along the Second Column
Expansion along \(C_2\) gives
\[|A|=a_{12}A_{12}+a_{22}A_{22}+a_{32}A_{32}\]
The signs in the second column are
\[-,\quad +,\quad -\]
Therefore,
\[\begin{aligned}|A|&=-a_{12}\begin{vmatrix}a_{21} & a_{23}\\a_{31} & a_{33}\end{vmatrix}+
a_{22}\begin{vmatrix}a_{11} & a_{13}\\a_{31} & a_{33}\end{vmatrix}\\&\quad-
a_{32}\begin{vmatrix}a_{11} & a_{13}\\a_{21} & a_{23}\end{vmatrix}\end{aligned}\]
Expansion Along the Third Column
Expansion along \(C_3\) gives
\[|A|=a_{13}A_{13}+a_{23}A_{23}+a_{33}A_{33}\]
The signs in the third column are
\[+,\quad -,\quad +\]
Hence,
\[\begin{aligned}|A|&=a_{13}\begin{vmatrix}a_{21} & a_{22}\\a_{31} & a_{32}\end{vmatrix}-
a_{23}\begin{vmatrix}a_{11} & a_{12}\\a_{31} & a_{32}\end{vmatrix}\\&\quad+
a_{33}\begin{vmatrix}a_{11} & a_{12}\\a_{21} & a_{22}\end{vmatrix}\end{aligned}\]
Six Equivalent Expansions of a \(3\times3\) Determinant
For a determinant of order \(3\), any one of its three rows or three columns may be selected for expansion.
| Expansion |
General Form |
Sign Pattern |
| First row \(R_1\) |
\(a_{11}A_{11}+a_{12}A_{12}+a_{13}A_{13}\) |
\(+,-,+\) |
| Second row \(R_2\) |
\(a_{21}A_{21}+a_{22}A_{22}+a_{23}A_{23}\) |
\(-,+,-\) |
| Third row \(R_3\) |
\(a_{31}A_{31}+a_{32}A_{32}+a_{33}A_{33}\) |
\(+,-,+\) |
| First column \(C_1\) |
\(a_{11}A_{11}+a_{21}A_{21}+a_{31}A_{31}\) |
\(+,-,+\) |
| Second column \(C_2\) |
\(a_{12}A_{12}+a_{22}A_{22}+a_{32}A_{32}\) |
\(-,+,-\) |
| Third column \(C_3\) |
\(a_{13}A_{13}+a_{23}A_{23}+a_{33}A_{33}\) |
\(+,-,+\) |
How to Choose the Best Row or Column for Expansion
Although every row and column gives the same determinant value, some choices make the calculation considerably shorter.
For examination problems, preferably choose a row or column containing:
- the largest number of zeros;
- simple numerical values;
- common factors;
- terms that simplify easily;
- repeated or related expressions that allow quick factorisation.
If a row contains two zeros, the determinant can often be evaluated using only one cofactor. This is a major time-saving strategy in CBSE and competitive examinations.
