"Describe all possible values of for which the term
occurs in the expansion of a determinant with co-efficient "
This question is on a sample paper of our linear algebra exam, which I'm studying for, but I don't really understand it. I assume it's to do with the expansion by Minors, but are these the co-ordinates of a single row or column and I'm being asked to find the values? Would someone mind explaining it to me please?? :)
Also, if you'd like to give me a hint as to how to start the question or something like that, I'd really appreciate it!
Originally Posted by Conn
Remember, or know, that the determinant of an n x n matrix is the sum of n! terms each with a definite sign and each being a product of n entries
from the matrix, products that contain exactly one element from each row in the matrix and exactly one element from each column in the matrix, so...
Ah yes, I see now, it's really more of a permutations question! So would I be right in saying (our lecturer doesn't post any solutions to questions) that since (i,j) = (5,7) or (7,5), and (k,l) = (2,6) or (6,2), there are four possible permutations of (i,j,k,l), two of which are even, and two are odd. The odd ones, which have co-efficient -1, are (i,j,k,l) = (5,7,6,2) and (7,5,2,6) by my calculations, and this is the answer??
Also, while we're on the subject of determinants, if you'd be so kind to help me again, we were given a n x n matrix with all diagonal entries equal to 0 and all off-diagonal entries equal to 1, and asked to find the determinant. I understand that the answer is , but I got this answer by finding the deteminant of the 2x2, 3x3, 4x4 and 5x5 matrices of similar form and then generalising the pattern. I'm not sure how to do a formal proof of the fact, however; I tried induction but it didn't really work, and I'm not sure how else to go about it.
Thanks again :)