1. ## Determinants

"Describe all possible values of $i, j, k, l$ for which the term

$a_{4k}a_{35}a_{il}a_{67}a_{j1}a_{23}a_{14}$

occurs in the expansion of a $7x7$ determinant with co-efficient $-1$"

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!

2. Originally Posted by Conn
"Describe all possible values of $i, j, k, l$ for which the term

$a_{4k}a_{35}a_{il}a_{67}a_{j1}a_{23}a_{14}$

occurs in the expansion of a $7x7$ determinant with co-efficient $-1$"

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!

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...

Tonio

3. 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 $(n-1).(-1)^{n+1}$, 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