1. ## Tricky equation

Hello,

There is a linear equation, in matrix form (like a system) giving me a hard time, this is it:

--.................--
|1, x, x^2, x^3 |
|x^3, 1, x, x^2 |
|x^2, x^3, 1, x | = 0
|x, x^2, x^3, 1 |
--.................--

(hope it looks like a matrix)

1. Am I allowed to solve only one line, it sounds stupid but they look pretty the same, does it make any difference that there are four lines, (I am asking this cause I solved the first one and it gives three values for x: -1, i, -i)

2. But I am not satisfied with this, there must be some theoreme, some topic I am missing, is it anything related to digonalization, inversion, eigen values-vectors, characteristic polynomial equation, please can someone give any idea what algebra topics I should study for this?

3. If someone could give a solution I would apreciate it a lot in order to compare it with mine (if I ever find one), I am so curious to know what is the solution.

4. They called it an "equation", I suppose because there is only one variable, x. Since they call it an equation why it is presented like a system?

5. I know how to use gaussian elemination, cramer etc, is this of any help or I should study cubic non linear systems (for which I have no idea so far)?

Thank you all,
Melsi

2. Hello, Melsi!

This is a rather silly problem.
They gave the same equation four times . . . why?

. . Can we write: . $\begin{array}{ccc}x^2 + 3x + 2 &=& 0 \\ x^2 + 2 + 3x &=& 0\\ 3x+ x^2 + 2&=& 0 \\ 3x + 2 + x^2 &=& 0 \\
2 + x^2+3x &=&0 \\ 2 + 3x +x^2 &=&0\end{array} \quad\hdots$
and say "Solve the system of equations" ??
I don't think so!

If they had used three different variables, say, $x,y,z$, it would make more sense.

$\left[\begin{array}{cccc}1 & x & x^2 & x^3 \\x^3& 1& x& x^2\\ x^2& x^3& 1& x \\ x& x^2& x^3& 1\end{array}\right] \;=\;\left[\begin{array}{c}0\\0\\0\\0\end{array}\right]$

Am I allowed to solve only one line? . . . . Yes!
I solved the first one and it gives three values for x: . $-1,\pm i$ . . . . Good!

3. ## I am wondering my self too!

Hello,

Thank you very much for your answer. The equation was part of a past Algebra examination-test taken at my University, the course is called "Linear Algebra". I am wondering my self, why did they ask such thing (in that course we learn about linear systems, digonalization, eigen values, subspaces, bases etc), how is this equation related to all these topics since it is so easy? Maybe I ask some assistant of the proff some day (the proff him self is too strange he may get mad if I ask him).

Anyway do you mean that the solution is the one I gave? Still wondering why in hell they ask that?

Thank you again my friend, it was kind of you,
Melsi

4. If I were to guess they probably meant the determinant... That is they asked you to solve:

$\left|\begin{array}{cccc}1 & x & x^2 & x^3 \\x^3& 1& x& x^2\\ x^2& x^3& 1& x \\ x& x^2& x^3& 1\end{array}\right|= 0$

I am sure you know what a determinant is..

5. Thank you Isomorphism, they asked solving this equation actually, anyway I too think it is a simple equation.

ancient greek: Isos-ίσος (which means equal)
ancient greek: Morphi-μορφή (shape,external shape, kind or type of something)

[or Morphie... not sure about English version]

[Isos + Morphi]>Isomorphos (shaped-formed in the same way)

os-ος: a common ending in modern Greek to form adjectives (omorphos=beautiful, amorphos=without shape, kalos, palios, mikros, megalos etc)

[Isomorphos]>Isomorphismos [a state where two things have the same form-shape]

mos: is a common ending to form nouns that describe-refer to some situation-state in modern greek (kataklysmos, katatregmos etc).

6. Originally Posted by Melsi
Thank you Isomorphism, they asked solving this equation actually, anyway I too think it is a simple equation.

ancient greek: Isos-ίσος (which means equal)
ancient greek: Morphi-μορφή (shape,external shape, kind or type of something)

[or Morphie... not sure about English version]

[Isos + Morphi]>Isomorphos (shaped-formed in the same way)

os-ος: a common ending in modern Greek to form adjectives (omorphos=beautiful, amorphos=without shape, kalos, palios, mikros, megalos etc)

[Isomorphos]>Isomorphismos [a state where two things have the same form-shape]

mos: is a common ending to form nouns that describe-refer to some situation-state in modern greek (kataklysmos, katatregmos etc).
Isomorphic - mathematical term derived from Greek meansing same structure

Isomorphism - a structure preserving mapping between Isomorphic structures.

RonL