Introduction:
In this problem, we are given a linear operator T on the space V of all n×n matrices. The operator T is defined as T(A) = A A^T/2, where A^T represents the transpose of matrix A. We are required to find the nullity of T.

Definition of Nullity:
The nullity of a linear operator is the dimension of its null space. The null space of a linear operator is the set of all vectors in the domain that map to the zero vector in the codomain.

Finding the Null Space of T:
To find the null space of T, we need to determine all matrices A such that T(A) = 0.

Let A be a matrix in V such that T(A) = 0. Then, we have A A^T/2 = 0. Multiplying both sides by 2, we get A A^T = 0.

Properties of Transpose:
The transpose of a matrix A satisfies the property (A^T)^T = A. Using this property, we can rewrite the equation as (A^T)^T A^T = 0.

Matrix Multiplication:
The product of two matrices A and B is the zero matrix if and only if either A or B is the zero matrix. Applying this property to our equation, we have (A^T)^T = 0 or A^T = 0.

Null Space:
For a matrix A, if A^T = 0, then all the entries of A are zero. This implies that A is the zero matrix.

Conclusion:
From the above analysis, we can conclude that the null space of the linear operator T consists only of the zero matrix. Therefore, the nullity of T is 0, as the dimension of the null space is 0.

Summary:
The nullity of the linear operator T, defined as T(A) = A A^T/2, is 0. This means that the null space of T consists only of the zero matrix.