# symbolic computation with any number of symbolic parameters. How to Simplify them?

I want to define d×d matrix $$A$$ as $$A_{mn} = \exp{[\sum_{i=0}^m\sum_{j=0}^n \theta_{ij}]}$$. Here $$\theta_{ij}$$ is real numbers. I wrote below script in Mathematica 12.

d = 3; Clear[A]; Clear[theta]
Array[A, {d, d}]; Array[theta, {d, d}];
For[m = 0, m < d, m++, {
For[n = 0, n < d, n++, {
A[m, n] = E[Sum[theta[i, j], {i, 0, m}, {j, 0, n}]]
}]}];
mtx = Table[A[i, j], {i, 0, d-1}, {j, 0, d-1}];
mtx // MatrixForm


The Purpose of my script is getting $$\rm{det}[A]$$.

Det[mtx]


How can I simplify it?

Det[mtx] // FullSimplify


I think I should use assumpution of $$\theta_{ij} \in \mathbb{R}$$. But I do not know how to do it.

• How could you write Array[A, {n, n}]; Array[theta, {n, n}]; when n is not defined? Array[A, {n, n}] gives error. You did not get an error on your Mathematica? Your sum is also wrong Sum[theta[i, j], {0, i, m}, {0, j, n}] you can't use zero as summation index. May be you meant Sum[theta[i, j], {i, 0, m}, {j, 0, n}] It is better to evaluate each code one by one to see the problems easily. You also have m++ { which is wrong. Commented Jun 17, 2020 at 2:10
• What exactly doesn't work?
– JimB
Commented Jun 17, 2020 at 2:13
• @Nasser Thank you for your comment. I fixed them. Commented Jun 17, 2020 at 3:12

d = 3;
Clear[A]; Clear[theta]
Array[A, {d, d}]; Array[theta, {d, d}];
For[m = 0, m < d, m++,
For[n = 0, n < d, n++,
A[m, n] = Exp[Sum[theta[i, j], {i, 0, m}, {j, 0, n}]]
]
];
(mat = Table[A[i, j], {i, 0, d - 1}, {j, 0, d - 1}]) // MatrixForm


Dimensions[mat]
(* {3, 3} *)

Det[mat] // Simplify