Definition of determinant for 2x2 matrix

[MathewsMD]

For a 2x2 matrix, does the general definition hold?

If so, how exactly is the minor $M_{i,j}$ computed in this case? If A is a 2x2 matrix, is det(A) only defined as ad - bc?

 

[Mark44]

For a 2x2 matrix, does the general definition hold?

If so, how exactly is the minor $M_{i,j}$ computed in this case? If A is a 2x2 matrix, is det(A) only defined as ad - bc?

The minor in this case is 1 x 1 matrix determinant. IOW, the number that isn't in row i, column j.

$$\det \begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc$$

 

[HallsofIvy]

For a 2x2 matrix, does the general definition hold?

If so, how exactly is the minor $M_{i,j}$ computed in this case? If A is a 2x2 matrix, is det(A) only defined as ad - bc?

No, that definition does not hold for any matrix because the determinant is a number where, as you have it, it is a function of "i".
What you should have is a specific number in place of "i". Typically it is "1" but any number from 1 to n would do.

In the case of a 2 by 2 matrix, n= 2 so, with your "i" replaced by "1", $\sum_{j= 1}^N (-1)^{1+ j}a_{1j}M_{1j}$ is $(-1)^{1+ 1}a_{11}a_{22}+ (-1)^{1+ 2}a_{12}a_{21}= a_{11}a_{22}- a_{12}a_{21}$. In the case that
$$\begin{bmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{bmatrix}= \begin{bmatrix}a & b \\ c & d \end{bmatrix}$$
that would be ad- bc.

Notice that if we replaced "1" above by "2" we would have $\sum_{j= 1}^N (-1)^{2+ j}a_{2j}M_{2j}$ is $(-1)^{2+ 1}a_{21}a_{12}+ (-1)^{2+ 2}a_{22}a_{11}= -a_{12}a_{21}+ a_{11}a_{22}$ the same as before.

The "minor", $M_{ij}$ is, by definition, the determinant of the matrix you have after removing the "ith row" and "jth column". In the case that
$$A= \begin{bmatrix} a & b \\ c & d\end{bmatrix}$$
$a_{11}= a$ and $M_{11}= a_{22}= d$ since we remove the first row, $\begin{bmatrix}a & b\end{bmatrix}$ and first column, $\begin{bmatrix}a \\ c \end{bmatrix}$ we are left with the "1 by 1 matrix", d.
Similarly, $a_{12}= b$ and $M_{12}= a_{21}= c$.