To find the value of the square of the determinant formed by its cofactors, we need to understand the concept of cofactors and how they are related to the original determinant.


The cofactor of an element in a determinant is the determinant formed by removing the row and column of that element. For a third-order determinant, each element has its own cofactor.


Given that the value of the third-order determinant is 11, we can represent it as follows:

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

The value of the determinant can be calculated using the formula:
det = aei + bfg + cdh - ceg - bdi - afh

Since the value of the determinant is 11, we have:
11 = aei + bfg + cdh - ceg - bdi - afh


To find the square of the determinant formed by the cofactors, we need to find the determinant of the cofactor matrix and then square it.

The cofactor matrix is formed by replacing each element of the original matrix with its cofactor. For example, the cofactor of 'a' is the determinant formed by removing the row and column of 'a'.

| e f |
| h i |

The determinant of this cofactor matrix can be calculated using the formula:
det(cofactor) = ei - fh

To find the square of this determinant, we simply square the value:
(det(cofactor))^2 = (ei - fh)^2


Let's substitute the values of the determinant and the cofactor into the equation:

(det(cofactor))^2 = (ei - fh)^2 = (11 - (bdi + afh + ceg - bfg - cdh))/a

Simplifying this expression, we get:
(det(cofactor))^2 = (11 - (bdi + afh + ceg - bfg - cdh))^2/a^2

Since we are given that the value of the determinant is 11, we can substitute this value into the equation:
(det(cofactor))^2 = (11 - (bdi + afh + ceg - bfg - cdh))^2/11^2

Simplifying further, we get:
(det(cofactor))^2 = (11 - (bdi + afh + ceg - bfg - cdh))^2/121

Therefore, the value of the square of the determinant formed by its cofactors is 11^2 = 121, which corresponds to option 'B'.