1. ## magic matrix

Hello

I would like to have help solving the problem:

I have to invent a marix 3*3 with entries from 1 to 8 such that all rows and all columns and al diagonals add to 12
the first row must be 7,2,3

thank you
i don't want the answer i want an explanation on how to get it

2. Originally Posted by qwerty321
Hello

I would like to have help solving the problem:

I have to invent a marix 3*3 with entries from 1 to 8 such that all rows and all columns and al diagonals add to 12
the first row must be 7,2,3

thank you
i don't want the answer i want an explanation on how to get it
I have a question. A $\displaystyle 3 \times 3$ matrix has nine entries, but you've only given us 8 numbers to work with??

3. 1- I gave u 9 entries (the 0 is included)
2- u can use the same entry twice

i have the answer but i need the methode on how to get it
do u think we can solve it by a system?

5. Originally Posted by qwerty321
i have the answer but i need the methode on how to get it
do u think we can solve it by a system?
Sure. If your matrix is $\displaystyle \left( \begin{array}{ccc} 7 & 2 & 3\\ a & b & c \\ d & e & f \\ \end{array}\right)$ then adding up rows, columns and diagonal gives the following system of equations

rows: $\displaystyle a + b + c = 12$, $\displaystyle d + e + f = 12$,
columns: $\displaystyle 7 + a + d = 12$, $\displaystyle 2 + b + e = 12$, $\displaystyle 3 + c + f = 12$,
diagonals: $\displaystyle 7 + b + f = 12$, $\displaystyle 3 + b + d = 12$

Then you can solve the system giving you the entries.

6. hehe no way u can solve this matrice..try it
look i got the answer and it seems that i don't need a proof of it..but what i need to prove is that:
is the matrix found unique?or is there other(s) matrices...so if u could solve the system u told me about,i think i can prove that it is unique

thank you

7. Originally Posted by qwerty321
hehe no way u can solve this matrice..try it
look i got the answer and it seems that i don't need a proof of it..but what i need to prove is that:
is the matrix found unique?or is there other(s) matrices...so if u could solve the system u told me about,i think i can prove that it is unique

thank you
The system I mentioned earlier has the solution

$\displaystyle a = 0,\; b =4,\; c = 8,\;d=5,\; e = 6,\; f = 1.$

This is the only solution so it is unique!

8. well that is correct
woow can u tell me how u solved the system?
thnk you
can u pm me?
i really than you

9. Originally Posted by qwerty321
well that is correct
woow can u tell me how u solved the system?
thnk you
can u pm me?
i really than you
Sure, the easiest way is to start with the system and solve equation by equation for a variable
Originally Posted by danny arrigo
$\displaystyle a + b + c = 12$, $\displaystyle d + e + f = 12$,
$\displaystyle 7 + a + d = 12$, $\displaystyle 2 + b + e = 12$, $\displaystyle 3 + c + f = 12$,
$\displaystyle 7 + b + f = 12$, $\displaystyle 3 + b + d = 12$
so from the first two

$\displaystyle c = 12-a-b$, $\displaystyle f = 12-d-e$, and keeping going like this solving for a new unknown (and substituting what you know). At the end you'll come up with a single equation for the last unknown and you'll know all the variables.

An interesting variation: Find all 3 X 3 matrices such that all the rows, columns and diagonals add up to the same number n and is this matrix unique?

10. ok i got a=5-d
b=10-e
c=9-f
b=5-f
b=9-d
but i can't see how to substitute?

11. Originally Posted by qwerty321
ok i got a=5-d
b=10-e
c=9-f (*)
b=5-f
b=9-d
but i can't see how to substitute?
The first and second are good $\displaystyle a = 5 - d, b = 10-e$ but in (*), you already have c. In fact, with c and f above, this equation is already satisfied! Continue picking ones that you haven't already picked.

12. hey it ain't workings
When I substitute i always get 2 variables

13. ok i got it
and it is not unique because we can rotate it right?

14. Originally Posted by qwerty321
ok i got it
and it is not unique because we can rotate it right?
Yes, that's right but if you specify the first row, then it is unique.