# Is cofactor matrix of 2x2 matrices unique?

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?

## 1 Answer

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.

• Exactly. What the first link calls the cofactor matrix is actually the adjugate matrix. – darij grinberg Feb 14 '20 at 10:39
• @darijgrinberg: I guess that there are other anomalies on that page. – Yves Daoust Feb 14 '20 at 10:44