Question

Evaluate the determinants to verify the equation

Answer 1

I have tried solving this and have been unsuccessful. Can anyone help me out?

Answer 2

is this your problem child of a problem?

Answer 3

lol my problem

Answer 4

ummm the equation after the equal sign. Is that the determinant?

Answer 5

Since i know what the determinant equals but it doesnt equal that

Answer 6

it's asking you to evaluate the determinant and then show it does equal the expression on the RHS (right and side).

Answer 7

well it doesnt LOL

Answer 8

sure? I find the determinant to be
\[ bc^3 + ca^3 + ab^3 - (ba^3 + cb^3 + ac^3) \]

Answer 9

ya i got that too

Answer 10

while the RHS,
\[ RHS = (-a^2b + a^2c + ab^2 - ac^2 - b^2c + bc^2)(a+b+c) \]
\[= ab^3 + bc^3 + ca^3 -(ba^3 - ac^3 - cb^3) \]
which is the same.

Answer 11

in the brackets, should be + +

Answer 12

LOL I dont know where I went wrong but when I foiled I Landed up with too many cs

Answer 13

cs?

Answer 14

like the variable c

Answer 15

This problem is training you. Time to move beyond FOIL and multiply things out and keep track of them.

Work this again and make sure you can replicate the result.

Answer 16

like what do u mean by moving beyond foil?

Answer 17

I mean you shouldn't need the rubric anymore. And what's more, you are seeing things with more than two terms in each bracket now. You should teach yourself to do that without using foil, a la the following

(x + y + z)(a + b + c)
= xa + xb + xc + ya + yb + yc + za + zb + zc

Answer 18

ya but like there it was (a-b)(b-c)(c-a)

Answer 19

so first i did (a-b)(b-c) and then i multiplied my answer by (c-a)
Is that correct?

Answer 20

exactly so, that's equal to
(ab - ac -b^2 + bc)(c-a)
= abc - ac^2 -b^2c + bc^2 -a^2b+a^2c + ab^2 - abc
= ab^2 + bc^2 + ca^2 - (ac^2 + ba^2 + ac^2)

Answer 21

etc., etc.

Lots of ways to skin this cat. But having the discipline to work through it is important.

Answer 22

okkkkk i just caught my mistake LOL

Answer 23

ok, I'm going to move on.

Answer 24

You make one silly mistake and it ruins everything. Thanks James for ur help

Answer 25

Yes. Here's a hint for this problem.

Clearly the problem is symmetric in a,b,c. Which means if we swapped the order of them it shouldn't upset the result too much. That being the case, at every step of the way, we should expect a certain symmetry in the terms.

So if there's a term such as
ab^2

it make sense that we should also expect a term such as
bc^2
and
ca^2

This is a way to at least sense check your answer at each step.

Answer 26

Alrighty Thanks for ur help :DDDDDDDD