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In linear algebra, the determinant is exchanged for any two rows. The determinant changes the sign once, so the two rows must be adjacent? What if it's a matrix? Do matrices use variable signs and why?
Determinant exchanges between rows and columns and columns do not have to be adjacent. The exchange of rows and columns of a matrix does not need to change the sign. After the exchange, it is equivalent to multiplying or multiplying an elementary matrix by the left or right . It is no longer the original matrix, but it is similar to the original matrix and has the same eigenvalues.
follow up
Multiplied by this elementary matrix is?
There is another one, if a row of the matrix is directly divided by two, the resulting matrix is still equal to the original matrix?
follow up
If you swap rows, then left-multiply an elementary matrix: this elementary matrix is a diagonal matrix I (denoted as E in some books, that is, a matrix with all 1s except the main diagonal) The corresponding rows are swapped.
For example, exchanging one or two rows:
1 2 3 4 5 6
4 5 6 → 1 2 3
7 8 9 7 8 9
is equivalent to the diagonal matrix I
4 5 6 0 1 0 1 2 3
1 after exchanging one or two rows by left multiplication 2 3 = 1 0 0 · 4 5 6
7 8 9 0 0 1 7 8 9
Swap columns to multiply this elementary matrix on the right, which is also the result of exchanging the corresponding columns of the diagonal matrix I.
After multiplying by two, it is no longer the original matrix.