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Let's assume I have a matrix $$\begin{pmatrix} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}= \begin{pmatrix} x \\ y \\ z \end{pmatrix}$$ assume $x = a_1+b_1+c_1,y=a_2+b_2+c_3,z = a_3+b_3+c_3$, then I calculate the trace and determinant of the matrix.

Now given the values of $(x,y,z)$, trace, determinant, the range of $a_1....... c_3$ is between $0-7$ and are integers.

What are my chances or any other ways for me to get back to the original matrix?

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    $\begingroup$ I don't understand how a $3 \times 3$ matrix can equal a $3 \times 1$ one. $\endgroup$
    – Randall
    Aug 30, 2021 at 17:19
  • $\begingroup$ Consider it as a system of equations or I write a function on a matrix which gives me values $(x,y,z)$. $\endgroup$
    – User1086
    Aug 30, 2021 at 17:22
  • $\begingroup$ But they're still not equal. They usually take the form $A \mathbf{x} = \mathbf{b}$, not $A = \mathbf{b}$. $\endgroup$
    – Randall
    Aug 30, 2021 at 17:23
  • $\begingroup$ You can write $$\begin{pmatrix} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}= \begin{pmatrix} x \\ y \\ z \end{pmatrix}$$ $\endgroup$
    – user
    Aug 30, 2021 at 17:24
  • $\begingroup$ This might be the correct way to write. $\endgroup$
    – User1086
    Aug 30, 2021 at 17:26

1 Answer 1

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We have $5$ equations with $9$ unkowns, that is

  • $a_1+b_1+c_1=x $
  • $a_2+b_2+c_3=y $
  • $a_3+b_3+c_3=z$
  • $tr(A)=a_1+b_2+c_3=u$
  • $\det(A)=v$

we could use the first $4$ linear equations as constraint and then search numerically for the solutions which satisfies the equation for the determinant for all the possible combination of the remaining $5$ free parameter, which means $8^5=2^{15}=32768$ trials.

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  • $\begingroup$ @ user, So it is possible for me to write an algorithm or some code in any programming language that can search for these combinations is that correct? $\endgroup$
    – User1086
    Aug 30, 2021 at 17:36
  • $\begingroup$ Is there any other value of the matrix that I can store so I reduce the possible solutions? $\endgroup$
    – User1086
    Aug 30, 2021 at 17:38
  • $\begingroup$ Yes we can proceed numerically and find the solution(s). $\endgroup$
    – user
    Aug 30, 2021 at 17:38
  • $\begingroup$ I don’t understand the second question, sorry. $\endgroup$
    – user
    Aug 30, 2021 at 17:45
  • $\begingroup$ Can I calculate any other values from the matrix other than trace and determinant, so that number of possible solutions will be less than 32768 $\endgroup$
    – User1086
    Aug 30, 2021 at 17:54

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