Value of a 3rd Order Determinant:
The value of a 3rd order determinant can be found by expanding it along any row or column. Let's assume we expand it along the first row:

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

The value of this determinant can be calculated as follows:

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

Value of the Determinant formed by replacing each element by its co-factor:
The co-factor of an element in a matrix is the determinant formed by removing the row and column containing that element. The value of the determinant formed by replacing each element by its co-factor can be calculated using the following steps:

1. Calculate the co-factor of each element in the original matrix.
2. Replace each element in the original matrix with its corresponding co-factor.
3. Calculate the value of the determinant formed by the new matrix.

Let's assume the original matrix is:

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

Calculating the Co-factor Matrix:
The co-factor matrix is formed by calculating the co-factor of each element in the original matrix. The co-factor of each element can be found by taking the determinant of the 2nd order sub-matrix formed by removing the row and column containing that element.

The co-factor matrix is:

| e -d f |
| -h g -i |
| h -g i |

Calculating the Determinant:
To find the value of the determinant formed by replacing each element with its co-factor, we need to calculate the determinant of the new matrix. We can expand it along any row or column.

Expanding along the first row, we get:

Det = e(gi - hf) - (-d)(hi - fg) + f(-g) - (-d)(hi - fg)

Simplifying this expression, we get:

Det = egi - ehf + dhi - dfg + f(-g) + d(hi - fg)

Det = egi - ehf + dhi - dfg - fg + dhi

Det = egi - ehf + 2dhi - dfg - fg

Conclusion:
The value of the determinant formed by replacing each element with its co-factor is given by the expression egi - ehf + 2dhi - dfg - fg. This value can be calculated using the co-factor matrix obtained by calculating the co-factor of each element in the original matrix.