Please consider the 2x2 matrix below:
$\left[\begin{array}{ccc} 1 & 2 \\ 3 & 4 \end{array}\right]$
According to the definition given here and here, the cofactor matrix becomes:
$\left[\begin{array}{ccc} a_{22} & -a_{12} \\ -a_{21} & a_{11} \end{array}\right] = \left[\begin{array}{ccc} 4 & -2 \\ -3 & 1\end{array}\right]$
However, when I follow the practice given here and here, I do obtain the following cofactor matrix, which is the transpose of the above:
$\left[\begin{array}{ccc} a_{22} & -a_{21} \\ -a_{12} & a_{11} \end{array}\right] = \left[\begin{array}{ccc} 4 & -3 \\ -2 & 1\end{array}\right]$
The difference arises from the off-diagonal locations of $a_{12}$ and $a_{21}$.
Are these two cofactors equivalent to each other in some way?
$\endgroup$1 Answer
$\begingroup$The first "here" link is wrong. The second one, contrary to your thinking, gives the correct interpretation.
The rule is simple: to obtain the minor/cofactor of any element, strike out the whole row and column that contain it. Hence it cannot contain the element self.
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