Solved

how to prove if a rank(A)<n, then det(A)=0. (A is a matrix)

Posted on 2004-04-12
2
506 Views
Last Modified: 2012-08-14
hi, all,
Does anybody know how to prove that for the nxn matrix, if rank(A)<n, then det(A)=0;
any help will be highly appreciated.
Chris
0
Comment
Question by:chrisguo
[X]
Welcome to Experts Exchange

Add your voice to the tech community where 5M+ people just like you are talking about what matters.

  • Help others & share knowledge
  • Earn cash & points
  • Learn & ask questions
2 Comments
 
LVL 84

Accepted Solution

by:
ozo earned 125 total points
ID: 10808463
If rank(A) < n then one of the rows or columns of the matrix is a linear combination of the other rows or columns
if one of the rows or columns of the matrix is a linear combination of the other rows or columns then the determinate of the matrix is 0
0
 

Author Comment

by:chrisguo
ID: 10810484
ozo,
thanks for the answer, I think I got the answer already.
Since the determinant of a matrix has such properties as follows.
1. any row or column of a matrix added to one row or column, the determinant will not change.
2. if matrix B is one row or column of a matrix A multiply a constant k, then the det(B)=k*det(A);
3. if any row or column of a matrix A is 0, then det(A)=0;
And if rank(A)<n, then we must have at least two rows or columns are linearly dependant, it means k*Ai+Aj=0(Ai and Aj are the vectors of matrix A).
So if we do this operation on matrix A to get a matrix B in which one vector is 0, then we can have det(A)=k*det(B)=0. Then we got it proved.
0

Featured Post

Technology Partners: We Want Your Opinion!

We value your feedback.

Take our survey and automatically be enter to win anyone of the following:
Yeti Cooler, Amazon eGift Card, and Movie eGift Card!

Question has a verified solution.

If you are experiencing a similar issue, please ask a related question

Suggested Solutions

Title # Comments Views Activity
Vertical and Horizontal Shift 4 144
Honda Vezel hybrid vs Honda Shuttle hybrid 2 330
Pre-Linear Regression Tests 2 60
how does a seed know what it is when planted? 10 70
Introduction On a scale of 1 to 10, how would you rate our Product? Many of us have answered that question time and time again. But only a few of us have had the pleasure of receiving a stack of the filled out surveys and being asked to do somethi…
This article seeks to propel the full implementation of geothermal power plants in Mexico as a renewable energy source.
Although Jacob Bernoulli (1654-1705) has been credited as the creator of "Binomial Distribution Table", Gottfried Leibniz (1646-1716) did his dissertation on the subject in 1666; Leibniz you may recall is the co-inventor of "Calculus" and beat Isaac…
Finds all prime numbers in a range requested and places them in a public primes() array. I've demostrated a template size of 30 (2 * 3 * 5) but larger templates can be built such 210  (2 * 3 * 5 * 7) or 2310  (2 * 3 * 5 * 7 * 11). The larger templa…

738 members asked questions and received personalized solutions in the past 7 days.

Join the community of 500,000 technology professionals and ask your questions.

Join & Ask a Question