And here is for multiplication:
[math]n\times C(x)_{r}\begin{Bmatrix} \cdots \end{Bmatrix},s(r_{p})[/math]
[math]\frac{d_{i}n}{d_{e}}[/math],[math]rn[/math]
The next set of operations are the simple of addition and subtraction.
[math]n+ C(x)_{r}\begin{Bmatrix} \cdots \end{Bmatrix},s(r_{p})[/math]
[math]\frac{d_{i}}{d_{e}}+\frac{nd_{e}}{d_{e}}[/math], [math]r+n[/math]
And then there is the subtraction of Raymond Arithmetic:
[math]n- C(x)_{r}\begin{Bmatrix} \cdots \end{Bmatrix},s(r_{p})[/math]
[math]\frac{d_{i}}{d_{e}}-\frac{nd_{e}}{d_{e}}[/math], [math]r-n[/math]
Now onto powers:
[math](C(x)_{r}\begin{Bmatrix} \cdots \end{Bmatrix}, s(r_{p}))^{n}[/math]
[math](\frac{d_{i}}{d_{e}})^{n}[/math], [math]r^{n}[/math]
And here is rooting Collatz-Matrix equations:
[math]\sqrt[n]{C(x)_{r}\begin{Bmatrix} \cdots \end{Bmatrix}, s(r_{p})}[/math]
[math]\sqrt[n]{\frac{d_{i}}{d_{e}}}[/math], [math]\sqrt[n]{r}[/math]
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 [math]d_{c} = -1[/math].
Here would be an example of a Collatz-Matrix equation to the 0th that is 5x5 and x equals 1.
[math]\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}[/math]
A Collatz-Matrix equation, like this, could be simplified to a one dimensional Collatz-Matrix equation:
[math]C(x)_{k}\begin{Bmatrix} x-1 & x+1 \end{Bmatrix},s(k_{p})[/math]
This would apply to all dimensions of Collatz-Matrix equations:
In this case, the variable [math]s_{p}[/math] 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:
[math]d_{i}xr-d_{e}r[/math]
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.