I have made up this "fun" problem.
The first row of a 3x3 matrix is $(a_{11} a_{12} a_{13})$. The next row consists of the variables $h, g$ and $c$ in any order. The last row are distinct non-zero fixed scalars, such as $2, 1, 7$ or $-4$ etc. They could be any integer.
Given an expression for $det(A)$ for example:
$(-c -2h)a_{11} -3ca_{13} + 3ha_{12} + 2ga_{13} + ga_{12}$
What is the probability that a matrix you construct meeting the conditions given and with this determinate will be the original matrix A?
Ans: 1
If the last two rows are sets of distinct non-zero integers. What is the probability that a matrix you construct meeting the conditions given will be the original matrix A?
Example $det(A) = -71a_{11} +13a_{12} +11a_{13}$ how many matrices could produce this?
(This one seems to be limited to the number of solutions to a Diophantine system, and it's not I type I can solve, unless there is another way...)
If we let the scalars be any real number will the set of possible matrices become infinite?