The original determinant is:
| a 1 1 a 12 a 13 |
|A2 1 A22 A23|
|A3 1 A32 A33|
Swap the first two lines to get a new determinant.
|A2 1 A22 A23|
| a 1 1 a 12 a 13 |
|A3 1 A32 A33|
N-order determinant is defined as the multiplication of all n elements in different rows and columns and the sum of all symbols.
There are two ways to get the sign:
Method 1:-1 to the power of a, where a is the reciprocal of the column number when the line numbers of the selected elements are arranged in natural order (from small to large).
Method 2:-1 to the power of b, where b is the reciprocal of the number of rows when the column numbers of the selected elements are arranged in natural order (from small to large).
In fact, there is a third method:-1 to the c power, where c is the reciprocal of the row number of the selected element+the reciprocal of the column number.
Here, we use the first method to extract elements from the first row to the last row of different columns (to ensure that the row number is a natural sequence).
For example, the obtained element is A 12 A2 1 A33, and its number before exchange is 2, 1, 3 (regardless of its reverse number).
However, it (because no matter how rows or columns are exchanged, elements that are not in the same row or column cannot be exchanged to the same row or column, so these elements can definitely be retrieved after the exchange). After the exchange, the column numbers are arranged as 1, 2, 3.
The order of the new column number is equivalent to the order of the original column number, and two elements are exchanged (corresponding to which two lines you exchange, here we exchange 12 lines, so the column number is exchanged in the order of 12 columns).
Because the exchange (on this certificate) has changed the reverse order number, the same element symbol is used, but the opposite is true.
I don't know if I made it clear. To put it bluntly, it is to change the serial number. Ask again if you don't understand.