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How many 3x3 matrices can you find

  1. May 20, 2013 #1
    1. The problem statement, all variables and given/known data

    j0fluJ9.png

    My instructor wants to know if there are finite or infinite amount of solutions

    2. Relevant equations
    Matrix Multiplication


    3. The attempt at a solution
    I pretty much turned A into a 3x3 matrix like this...
    | A11 A12 A13 |
    | A21 A22 A23 |
    | A31 A32 A33 |

    and then multiplied it by the given matrix of x y and z.

    After multiplying I found myself with 2 matrices that I think would fit the solution...

    | y 0 0 |
    | 0 0 0 |
    | 0 0 0 |

    and

    | 0 x 0 |
    | 0 0 0 |
    | 0 0 0 |

    The first matrix would correspond to xy + 0y + 0z
    0x + 0y + 0z
    0x + 0y + 0z
    yielding the answer indicated by the question


    and the second would correspond to 0x + xy + 0z
    0x + 0y + 0z
    0x + 0y + 0z
    yielding the answered indicated by the question

    However, I find myself second guessing myself. I think I have 2 solutions, but since x and y are variables, they could be any number, ergo have infinite matrices. I think I may be over complicating things or perhaps I am just incorrect. Pretty much my question is, am I correct or perhaps on the right track or just wrong?
     
  2. jcsd
  3. May 20, 2013 #2

    DrClaude

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    Staff: Mentor

    This indeed doesn't work. You have to find a matrix that will work with any vector, and return a vector with only the first component non-zero, such that this component is the product of the first two components of the original vector.
     
  4. May 20, 2013 #3

    haruspex

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    Science Advisor
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    Gold Member
    2016 Award

    Consider some specific choice(s) of x and y.
     
  5. May 20, 2013 #4
    So working off of:

    A11x + A12y + A13z
    A21x + A22y + A23z
    A31x + A32y + A33z

    I should only be concerned with the first row? And in that first row I must somehow make it so it equals the desired result of xy? I feel like I've hit a brick wall on this problem. I do not understand how get from addition to multiplication. I have tried substituting random values for x and y trying to factor them out, but I feel like I am just shooting in the dark with that. Am I supposed to try to find some factor values and work off of those? I apologize if I am not grasping this quite well, this matrix stuff is new to me.
     
  6. May 20, 2013 #5

    Dick

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    Homework Helper

    Yes, just think about the first row. The entries of the matrix A have to be definite numbers, not variables like x or y I would think. Have you considered that there may be no such matrix A? Or did that say that the matrix A is allowed to contain the variables x,y and z?
     
    Last edited: May 20, 2013
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