The following forms the matrix b:

```
A := Simplify(D((Sqrt(2)*k*Pi^2*x^2*(x + y + z)*(x + y + z + j + m))/((x + y)*(x + y + z + j)*Sqrt(x + y + z + j + m + n)), {{k, x, y, z, j, m, n}}))
B := Simplify(D((Sqrt(2)*k*Pi^2*(x + y + z)*z^2*(y + z + j)*(x + y + z + j + m)*(z + j + m)*Sqrt(y + z + j + m + n))/((y + z)*(x + y + z + j)*(z + j)*(y + z + j + m)*Sqrt((x + y + z + j + m + n)*(z + j + m + n))), {{k, x, y, z, j, m, n}}))
F := Simplify(D((Sqrt(2)*k*Pi^2*(x + y + z + j + m)*(z + j + m)*m^2*Sqrt(((y + z + j + m + n)*(j + m + n))/((x + y + z + j + m + n)*(z + j + m + n)*(m + n))))/((y + z + j + m)*(j + m)), {{k, x, y, z, j, m, n}}))
G := Simplify(D(-((2*(-1 + k)*Pi)/k)/(8*Pi), {{k, x, y, z, j, m, n}}))
H := Simplify(D(-(2*Pi*(1 - ((x + y)*(y + z)*(x + y + z + j)*(y + z + j + m)*Sqrt((x + y + z + j + m + n)/(y + z + j + m + n)))/(k*y*(x + y + z)*(y + z + j)*(x + y + z + j + m))))/(8*Pi), {{k, x, y, z, j, m, n}}))
J := Simplify(D(-(2*Pi*(1 - ((x + y + z + j)*(z + j)*(y + z + j + m)*(j + m)*Sqrt(((x + y + z + j + m + n)*(z + j + m + n))/((y + z + j + m + n)*(j + m + n))))/(k*(y + z + j)*j*(x + y + z + j + m)*(z + j + m))))/(8*Pi), {{k, x, y, z, j, m, n}}))
K := Simplify(D(-(2*Pi - (2*Pi*Sqrt((x + y + z + j + m + n)*(z + j + m + n)*(m + n)))/(k*Sqrt((y + z + j + m + n)*(j + m + n)*n)))/(8*Pi), {{k, x, y, z, j, m, n}}))
b := {A, B, F, G, H, J, K}
```

Now, I need the 2nd, 4th and 6th elements of the first column of the inverted matrix. Instead of manually finding each component, I opted to use:

```
f := LinearSolve(b, {1, 0, 0, 0, 0, 0, 0})
```

Next, I used:

```
f((2)) + f((4)) + f((6))
```

to extract the required elements.

Here comes the part I have problems with. In my attempt to simplify the result of the actual sum I want, I used:

```
$Assumptions = Element(k, PositiveReals) && z1 <= z2 <= z3 <= z4 <= z5 <= z6 <= z7 && Element(z1, Reals) && Element(z2, Reals) && Element(z3, Reals) && Element(z4, Reals) && Element(z5, Reals) && Element(z6, Reals) && Element(z7, Reals)
TimeConstrained(Simplify(Together(((3*Pi)/4)*(f((2)) + f((4)) + f((6))) //. {x -> z2 - z1, y -> z3 - z2, z -> z4 - z3, j -> z5 - z4, m -> z6 - z5, n -> z7 - z6})), 3600)
```

Even with an hour, Mathematica couldn’t simplify the result.

The result I want is:

```
((-z1 + z3)*(-z1 + z5)*Sqrt(-z1 + z7))/(2*Pi*Sqrt(2)*k*(-z1 + z2)*(-z1 + z4)*(-z1 + z6))
```

How should I proceed?