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April 5th, 2013, 08:38 AM  #1 
Newbie Joined: Dec 2012 Posts: 17 Thanks: 0  Nontrivial solution of a homogeneous system
Hello, I'm stuck with this question. Can anyone help me? Find all values of k for which this homogeneous system has nontrivial solutions: [kx + 5y + 3z = 0 [5x + y  z = 0 [kx + 2y + z = 0 I made the matrix, but I don't really know which Gausselimination method I should use to get the result. Thanks already! 
April 5th, 2013, 12:45 PM  #2 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  Re: Nontrivial solution of a homogeneous system 
April 5th, 2013, 12:48 PM  #3 
Global Moderator Joined: Dec 2006 Posts: 20,469 Thanks: 2038 
The determinant of the coefficients matrix is 5(1  k), and you need it to be zero, so k = 1. Alternatively, multiply the first equation by 3 and add the second equation to it, then compare the result with the third equation. 
April 5th, 2013, 12:59 PM  #4 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  Re: Nontrivial solution of a homogeneous system
$\color{#003399}{\bf{skipjack}}$, as a moderator you should put your post ahead of mine because I think it will help the OP to read your solution first since it's short and sweet. Last edited by skipjack; June 20th, 2016 at 08:39 AM. 
April 5th, 2013, 10:40 PM  #5  
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  Re: Nontrivial solution of a homogeneous system
OK I managed to reorder posts with $\color{#003399}{\bf{skipjack}}$'s suggestion in PM. I still want to post my work even though it's longwinded as I believe it to use valuable ideas from Linalgebra. Quote:
Last edited by skipjack; June 20th, 2016 at 08:42 AM.  
June 19th, 2016, 10:20 PM  #6 
Newbie Joined: Jun 2016 From: INDIA Posts: 3 Thanks: 0 
I also have a same problem for 1.73a+d=0 1.73b+d=0 1.73c+d=0 a+b+c1.73d=0 How can I GET VALUES OF a,b,c,d 
June 20th, 2016, 08:57 AM  #7 
Global Moderator Joined: Dec 2006 Posts: 20,469 Thanks: 2038 
Is 1.73 an exact value or could it be exactly √3?

June 20th, 2016, 09:27 AM  #8 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,635 Thanks: 2620 Math Focus: Mainly analysis and algebra 
Trivially, $a=b=c$, so it's now a system of two variables

June 20th, 2016, 05:51 PM  #9 
Newbie Joined: Jun 2016 From: Kolkata Posts: 2 Thanks: 1 
Form the coefficient matrix and then solve for the values, but it might be a bit tricky for a 4*4 matrix

June 20th, 2016, 10:09 PM  #10 
Newbie Joined: Jun 2016 From: INDIA Posts: 3 Thanks: 0  

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homogeneous, nontrivial, solution, system 
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