What are the chances of recovering the matrix for the given assumptions.

Question

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?

Answer 1

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.