1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Stuck on problem with determinants

  1. Aug 9, 2006 #1
    hi all,

    have a rather sticky problem on determinants to deal with...hope someone can offer some help.

    [In what follows, the numbers in brackets denote suffixes, so that, for example, A(s)(j) refers to the element in the sth row of A and jth column of A.]

    Let A be an n x n matrix. Let s be a fixed integer with 1=<s=<n. Suppose that A(s)(j) = b(s)(j) + c(s)(j) for j = 1, 2, ... n. Let B and C be the n x n matrixes defined by B(i)(j) = C(i)(j) = A(i)(j) for j = 1, 2, ..., n, if i is not equal to s; B(s)(j) = b(s)(j), j = 1, 2, ..., n; and C(s)(j) = c(s)(j), j = 1, 2, ..., n. Then prove that det (A) = det (B) + det (C).

    First, we let c(s)(1), c(s)(2), ..., c(s)(n) be fixed and let d(B) = det (A) - det (C), where the rows of B, A, and C are the same except for row s; C(s)(j) = c(s)(j), j = 1, 2, ... n; A(s)(j) = B(s)(j) + C(s)(j), j = 1,2 , ...n.
    The main difficulty I face is in proving that d(B) satisfies the properties of a determinant function without assuming the expansion formula for a determinant. The properties in question are:

    1) If A' is obtained from A by an interchange of two rows of A, then d(A') = -d(A).

    2) If A' is obtained from A by multiplying one row of A by a real number p, then d(A') = p d(A).

    3) If A' is obtained from A by multiplying one row of A by a number and adding to another row, then d(A') = d(A).

    4) d(I) = 1.

    That d(B) satisfies these properties is fairly clear if row s is not involved. [for example, to prove d(b) satisfies 1): if we exchange two rows of B, neither of which is row s, this involves an interchanging of two rows of A and two rows of C, giving d(B') = -det(A) + det(C) = -d(B).]

    but what if we consider the case when row s is interchanged with another row...?? I can't think of any way other than to expand the determinants of A' and C'! hope someone can help me out here..thanks.
    Last edited: Aug 10, 2006
  2. jcsd
  3. Aug 10, 2006 #2
    Suppose you row-reduce all the other rows (not s) in the three matrices (A, B, and C) to echelon form using the same sequence of row operations (just sort of ignore the s row as this happens). Notice that the these rows are still equal among the matrices (the i-th row of A is the same as the i-th row of B, etc.). Then do necessary row exchanges to slide the s-th row into k-th row, where k is the column (or one of the columns) that lacks a pivot position. So each matrix now has been left multiplied by some matrix E (made from elementary ones). Then do some "multiplying one row of A by a number and adding to another row" till that special s row has 0's up till the k-th entry. Then consider the diagonal determinants.

    And now you can fill in the blanks. Maybe there's a better way to do this problem, but I don't know enough algebra. :smile: Or maybe I do, and I'm just not seeing it....
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Stuck on problem with determinants
  1. Stuck on problem (Replies: 1)

  2. Stuck on this problem (Replies: 11)