Given that p+q+r=0, solve the following equations for x: 

p-x r q
r q-x p
q p r-x

the entire matrix should be set equal to 0. (I guess this means that its determinant is 0)

in Calculus Answers by

Your answer

Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
To avoid this verification in future, please log in or register.

1 Answer


If we assume that, not only is the determinant zero, but that the components of the determinant give rise to three equations that equate to zero, then we have four equations and four unknowns. This implies we have sufficient information to find all four unknowns.

The determinant is:


If each of the square-bracketed expressions is zero then the determinant will also be zero. So:-

(q-x)(r-x)-p²=0 ①

r(r-x)-pq=0 ② 

pr-q(q-x)=0 ③ 

Also r=-(p+q). So ③ becomes -p(p+q)-q(q-x)=0, -p²-pq-q²+qx=0, x=(p²+pq+q²)/q.

And ② becomes r²-rx-pq=0, (p+q)²+(p+q)(p²+pq+q²)/q-pq=0.


The quadratic has complex roots, so choose 1+(p+q)/q=0 as the better option.

This means p+2q=0 and p=-2q. Therefore x=(4q²-2q²+q²)/q=3q.

Finally r=-(p+q)=q.

Now substitute these values for p, r and x into each equation:

(-2q)(-2q)-4q²=0 is true

q(-2q)+2q²=0 is true

-2q²-q(-2q)=0 is true

There are many solutions because there is no unique value for q.

So q is arbitrary, and p=-2q, r=q and x=3q. So the matrix can be rewritten:

-5q q q
q -2q -2q
q -2q -2q

It can be easily seen that the determinant is zero. The common factor can be taken outside the matrix and the matrix simply becomes the coefficients -5, 1 and -2.


by Top Rated User (616k points)

Related questions

0 answers
asked Jul 20, 2012 in Algebra 1 Answers by thinkofer Level 1 User (120 points) | 355 views
1 answer
asked May 26, 2018 in Pre-Algebra Answers by Subarna Das Level 1 User (440 points) | 41 views
Welcome to, where students, teachers and math enthusiasts can ask and answer any math question. Get help and answers to any math problem including algebra, trigonometry, geometry, calculus, trigonometry, fractions, solving expression, simplifying expressions and more. Get answers to math questions. Help is always 100% free!
82,257 questions
86,780 answers
3,660 users