Question

if a,b,c are positive and unequal.show that the value of the given determinant is negative.

Answer 1

I a b c I
I b c a I
I c a b I

Answer 2

Let's find the determinant
\[\Delta=a(bc-a^2)-b(b^2-ac)+c(ab-c^2) \]
Let's simplify it
\[abc-a^3-b^3+abc+abc-c^3\]
We get
\[3abc-a^3-b^3-c^3\]

Answer 3

Let me think, how we can prove this to be always negative!!

Answer 4

okay..

Answer 5

ash look -a^3 = always negative
-3^3 = -27
so i think its proved :)

Answer 6

That's correct but we need to prove that
\[abc<a^3+b^3+c^3\]
for all a's, b's and c's

Answer 7

take an example
|1 2 3|
|3 4 1|
|4 2 1|

Answer 8

we need to find its negative value and thats with a prove.

Answer 9

No, just one example is not enough, we need to prove it for all real numbers

Answer 10

yah .. @ash2326 is right @annas guys help me plz

Answer 11

yeah i m trying

Answer 12

plz ..

Answer 13

http://www.scribd.com/doc/88441256/19/Applications-of-Determinants-and-Matrices
check the example number 30

Answer 14

we'll have to perform little row and column operations

Answer 15

taht link is taking too much time ..

Answer 16

its a bit heavy site

Answer 17

@shruti if you want i can put a screen shot of the solution ?

Answer 18

yes sure. i think @ash2326 left.

Answer 19

ok give me a sec

Answer 20

fine

Answer 21

I'm here, I tried something but it wasn't correct :'(

Answer 22

:(aww...

Answer 23

check the attachment

Answer 24

no worries ash :) you did the best help you are the most awesome helper