hello,
please help me with this question


If the determinant of this matrix
x x^2 1+x^3
y y^2 1+y^3
z z^2 1+z^3
equals zero , then prove that xyz = -1, where x y and z are unequal.


I tried solving it directly but I couldn't end with xyz so i can prove it.
thanks .

Use elementary row operations. If you add a multiple of one row to another row then the determinant is unchanged. So subtract row 1 from row 2 and from row 3, and you get



Now you know that y–x is not zero, so you can divide through row 2 by y–x; and you can divide through row 3 by z–x. That gives you

Continue to simplify the determinant by similar manoeuvres. For example, you could now subtract row 2 from row 3, and subtract x times row 2 from row 1. By then, the determinant should look simple enough that you can work it out completely.

__________________

To view links or images in signatures your post count must be 10 or greater. You currently have 0 posts.

Asked here: Determine the value of (1 + xyz)

Thread closed.

__________________
There are two things you should never try to prove: the impossible and the obvious.

The greater danger for most of us lies not in setting our aim too high and falling short; but in setting our aim too low and achieving our mark. (Michelangelo Buonarroti)

• 
  To view links or images in signatures your post count must be 10 or greater. You currently have 0 posts.
  
• 
  To view links or images in signatures your post count must be 10 or greater. You currently have 0 posts.
  
• 
  To view links or images in signatures your post count must be 10 or greater. You currently have 0 posts.
  
• 
  To view links or images in signatures your post count must be 10 or greater. You currently have 0 posts.
  
• 
  To view links or images in signatures your post count must be 10 or greater. You currently have 0 posts.