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Math Help - magic matrix

  1. #1
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    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
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  2. #2
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    Quote Originally Posted by qwerty321 View Post
    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 3 \times 3 matrix has nine entries, but you've only given us 8 numbers to work with??
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  3. #3
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    1- I gave u 9 entries (the 0 is included)
    2- u can use the same entry twice
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  4. #4
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    come on please
    i have the answer but i need the methode on how to get it
    do u think we can solve it by a system?
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  5. #5
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    Quote Originally Posted by qwerty321 View Post
    come on please
    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 \left( \begin{array}{ccc}<br />
7 & 2 & 3\\<br />
a & b & c \\<br />
d & e & f \\<br />
\end{array}\right) then adding up rows, columns and diagonal gives the following system of equations

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

    Then you can solve the system giving you the entries.
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  6. #6
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    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
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  7. #7
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    Quote Originally Posted by qwerty321 View Post
    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

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

    This is the only solution so it is unique!
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  8. #8
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    well that is correct
    woow can u tell me how u solved the system?
    thnk you
    can u pm me?
    i really than you
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  9. #9
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    Quote Originally Posted by qwerty321 View Post
    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
    Quote Originally Posted by danny arrigo View Post
    a + b + c = 12, d + e + f = 12,
    7 + a + d = 12, 2 + b + e = 12, 3 + c + f = 12,
    7 + b + f = 12, 3 + b + d = 12
    so from the first two

    c = 12-a-b, 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?
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  10. #10
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    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?
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  11. #11
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    Quote Originally Posted by qwerty321 View Post
    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 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.
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  12. #12
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    hey it ain't workings
    When I substitute i always get 2 variables
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  13. #13
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    ok i got it
    and it is not unique because we can rotate it right?
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  14. #14
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    Quote Originally Posted by qwerty321 View Post
    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.
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