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

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?

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

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.

• @ 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? Aug 30, 2021 at 17:36
• Is there any other value of the matrix that I can store so I reduce the possible solutions? Aug 30, 2021 at 17:38
• Yes we can proceed numerically and find the solution(s).
– user
Aug 30, 2021 at 17:38
• I don’t understand the second question, sorry.
– user
Aug 30, 2021 at 17:45
• 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 Aug 30, 2021 at 17:54