GATE CSE
First time here? Checkout the FAQ!
x
+3 votes
788 views

If the following system has non-trivial solution, 

$px + qy + rz = 0$

$qx + ry + pz = 0$

$rx + py + qz = 0$,

then which one of the following options is TRUE?

 

  1. $p - q + r = 0 \text{ or } p = q = -r$
  2. $p + q - r = 0 \text{ or } p = -q = r$
  3. $p + q + r = 0 \text{ or } p = q = r$
  4. $p - q + r = 0 \text{ or } p = -q = -r$
asked in Linear Algebra by Veteran (76k points)   | 788 views

2 Answers

+12 votes
Best answer

for non-trivial solution $$\left | A \right | = 0$$

where $\left | A \right | = \begin{bmatrix} p & q& r\\ q& r& p\\ r& p & q \end{bmatrix} = p*(rq-p^{2})-q*(q^{2}-pr)+r*(qp-r^{2})$

$=prq - p^3 - q^3 + prq + prq - r^3 \\= 3prq - p^3 - q^3 - r^3 \\=-{\left(p+q+r\right)}^3 + 3(p+q+r)(pq+qr+pr)$

now if you check the options the only options where each individual condition can make $\left | A \right | = 0$ zero is C

answered by Active (2.2k points)  
selected by
nice edit @arjun sir +1 for the edit.....

for a homogeneous equation to have a consistent solution, the general equation:

$AX=0$

must be satisfied.
If $A^{-1}$ exists then we can multiply by $A^{-1}$ on both sides and get $X=0$, which means solution is trivial.

But if $A^{-1}$ does not exist, meaning $|A|=0$, we cannot multiply both sides by $A^{-1}$ to reach $X=0$. In which case it implies that other non-trivial solutions exists.

http://math.stackexchange.com/questions/1012571/non-trivial-solutions-for-homogeneous-equations

+8 votes
Answer = C

If we add all the equations we get

(p+q+r)x + (p+q+r)y + (p+q+r)z = 0

which implies p+q+r=0

Only option C has p+q+r=0
answered by Active (1.2k points)  
Answer:

Related questions



Top Users Mar 2017
  1. rude

    4018 Points

  2. sh!va

    2994 Points

  3. Rahul Jain25

    2804 Points

  4. Kapil

    2608 Points

  5. Debashish Deka

    2104 Points

  6. 2018

    1414 Points

  7. Vignesh Sekar

    1336 Points

  8. Bikram

    1218 Points

  9. Akriti sood

    1186 Points

  10. Sanjay Sharma

    1016 Points

Monthly Topper: Rs. 500 gift card

21,445 questions
26,757 answers
60,936 comments
22,954 users