in determinant expansion why we use + - + simultaneously
Answers
A row operation corresponds to multiplying a matrix A on the left by one of several elementary matrices whose determinants are easy to compute to get a matrix B=EA. For instance, swapping the rows of a 2x2 matrix is done with
(0110)(acbd)
The determinant of the resulting row-swapped matrix is the product of the two determinants. Hence detB=detEdetA. Since detE=−1 in this case, you can compute the det of the new matrix and multiply it be −1 to get the det of the original one. (This is typically not very useful, but it's an example).
In the same way, a column op is done with A↦AE, and you can use the same rule -- prduct of determinants -- to relate the determinant of B=AE to the determinant of A.
In short: you can do a sequence of row and column ops, each of which adds a factor to the determinant, until you reach the identity. You don't have to do just a sequence of row ops or just a sequence of column ops.
Personal advice: Just use one or the other. It'll take a little longer, but you're much less likely to make a mistake in my experience with many students over the years.