Cofactors
- Cofactor is simply the minor of each element together with its “place sign“.
- The cofactor is denoted by Cij.
- ‘i’ is the row number while ‘j’ is column number.
- When a matrix is built with all the cofactors of all the elements, it is known as a cofactor matrix.
The place sign
- Place signs are plus (+) or minus (-) symbols used to indicate the positions of elements within a matrix.
- The top left-hand element of the matrix always has a plus (+) place sign.
- And then other elements has alternately plus (+) or minus (-) place signs.
- The following is the place sign matrix of a 3 × 3 matrix.
\[\Large
\begin{bmatrix}
+ & – & + \\
– & + & – \\
+ & – & +
\end{bmatrix}
\]
The cofactor
- So when the minor and the place sign of a particular element is known, we can easily calculate the cofactor of that element.
\[\Large
A =
\begin{bmatrix}
2 & 1 & 4 \\
1 & 6 & 3 \\
5 & 2 & 2
\end{bmatrix}
\]
- Let’s take the element ‘1’ (a21) of the matrix ‘A’ to calculate the cofactor.
\[\large
M_{21} = -6 ~~~ \text{(Minor, calculated previously)}
\]
\[\large \text{Place sign is ‘-‘}
\]
\[\large
C_{21} = -(-6) = \mathbf{6}
\]
- Alternatively, the cofactor can be calculated using the following formula too.
\[\Large
C_{ij} = (-1)^{i+j}~M_{ij}
\]
- In the above formula,
- Cij – Cofactor.
- i – Row number.
- j – Column number
- Mij – Minor
What is the place value of the element “b” of the following matrix.
\[\Large
A =
\begin{bmatrix}
a & \color{blue}{b} & c \\
d & e & f \\
g & h & i
\end{bmatrix}
\]