And here is for multiplication:
n\times C(x)_{r}\begin{Bmatrix} \cdots \end{Bmatrix},s(r_{p})
\frac{d_{i}n}{d_{e}},rn
The next set of operations are the simple of addition and subtraction.
n+ C(x)_{r}\begin{Bmatrix} \cdots \end{Bmatrix},s(r_{p})
\frac{d_{i}}{d_{e}}+\frac{nd_{e}}{d_{e}}, r+n
And then there is the subtraction of Raymond Arithmetic:
n- C(x)_{r}\begin{Bmatrix} \cdots \end{Bmatrix},s(r_{p})
\frac{d_{i}}{d_{e}}-\frac{nd_{e}}{d_{e}}, r-n
Now onto powers:
(C(x)_{r}\begin{Bmatrix} \cdots \end{Bmatrix}, s(r_{p}))^{n}
(\frac{d_{i}}{d_{e}})^{n}, r^{n}
And here is rooting Collatz-Matrix equations:
\sqrt[n]{C(x)_{r}\begin{Bmatrix} \cdots \end{Bmatrix}, s(r_{p})}
\sqrt[n]{\frac{d_{i}}{d_{e}}}, \sqrt[n]{r}
Here are a few properties to notice.
For multiplication, you may never divide or multiply by 0.
For exponentiation, where n is equal to 0...
Where d_{c} = -1.
Here would be an example of a Collatz-Matrix equation to the 0th that is 5x5 and x equals 1.
\begin{bmatrix} 1 & 2 & 3 & 4 & 5\\ 0 & 0& 0& 0 & 5\\ 0 & 0& 0& 0 &5 \\ 0& 0& 0& 0&5 \\ 0 & 0 & 0& 0 &5 \end{bmatrix}
A Collatz-Matrix equation, like this, could be simplified to a one dimensional Collatz-Matrix equation:
C(x)_{k}\begin{Bmatrix} x-1 & x+1 \end{Bmatrix},s(k_{p})
This would apply to all dimensions of Collatz-Matrix equations:
In this case, the variable s_{p} is a special case dimension, which refers to this Collatz-Matrix equation as a trans-linear equation.
It is not proven yet, but I postulate that the if for any determinant of a Collatz-Matrix equation if solving for x when y equals -1, x will always equal 1.
A formula that is used for finding the numbers that will work in all directions in a matrix solution for Collatz-Matrix equations is the following:
d_{i}xr-d_{e}r
As a side note, if a parameter is simply equal to x, then it is not necessary to include it within the notation. However, if you are determining the determinant of the Collatz-Matrix equation it will be necessary to include it within the parameter brackets to show the existence of these parameters.