**Determinant of a 3rd Order Matrix**
The determinant of a 3rd order matrix can be calculated using the formula:

det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)

Where A is the 3x3 matrix:

| a b c |
| d e f |
| g h i |

**Determinant of Co-factors**
The co-factor of each element in the matrix is calculated by taking the determinant of the 2x2 submatrix formed by excluding the row and column of the element. The co-factor is then multiplied by (-1)^(i+j), where i is the row index and j is the column index of the element.

The determinant of the matrix formed by the co-factors can be calculated using the formula:

det(cof(A)) = a1(a2i - a2j) - b1(b2i - b2j) + c1(c2i - c2j)

Where cof(A) is the matrix formed by the co-factors and a1, b1, c1 are the elements of the first row of A, and a2i, a2j, b2i, b2j, c2i, c2j are the co-factors of the remaining elements.

**Calculating the Value of det^2**
To find the value of det^2, we need to calculate the determinant of the matrix formed by the co-factors and then square it.

Let's say the determinant of the 3rd order matrix A is 9. We can use this information to find the determinant of the matrix formed by the co-factors, det(cof(A)).

Once we have det(cof(A)), we can square it to find the value of det^2.

**Conclusion**
In summary, the value of det^2, where det is the determinant formed by the co-factors of the elements of a 3rd order matrix, can be found by calculating the determinant of the matrix formed by the co-factors and then squaring it. The determinant of a 3rd order matrix can be calculated using the formula, and the co-factors are calculated by taking the determinants of the 2x2 submatrices formed by excluding the row and column of each element.