Rank of a Matrix:
The rank of a matrix is defined as the maximum number of linearly independent rows or columns in the matrix. It can also be interpreted as the dimension of the vector space spanned by the rows or columns of the matrix.

Rank of a Transpose:
The rank of the transpose of a matrix is equal to the rank of the original matrix. This means that if the original matrix has rank r, then the transpose of the matrix will also have rank r.

Given Information:
We are given a real 3 x 4 matrix A of rank 2.

Solution:
To find the rank of AtA, we need to consider the product of the transpose of A with A.

Let's calculate AtA:
- The matrix A has dimensions 3 x 4, so the transpose of A, At, will have dimensions 4 x 3.
- Multiplying At with A will result in a 4 x 4 matrix.

Since A is a 3 x 4 matrix of rank 2, it means that there are only two linearly independent rows or columns in A. This implies that the nullity of A is 2 (4 - 2 = 2).

Rank-Nullity Theorem:
According to the Rank-Nullity Theorem, for any matrix A, the rank of A plus the nullity of A equals the number of columns of A.

In this case, the number of columns of A is 4, and the nullity of A is 2. Therefore, the rank of A is 4 - 2 = 2.

Rank of AtA:
Since the rank of A is 2, it means that the rank of At (transpose of A) is also 2.

Now, consider the product AtA:
- The matrix At has dimensions 4 x 3, and A has dimensions 3 x 4.
- Multiplying At with A will result in a 4 x 4 matrix.

Since the rank of At is 2, it means that there are only two linearly independent rows or columns in At. This implies that the nullity of At is 2 (4 - 2 = 2).

According to the Rank-Nullity Theorem, the rank of At plus the nullity of At equals the number of columns of At.
- The number of columns of At is 4.
- The nullity of At is 2.

Therefore, the rank of At is 4 - 2 = 2.

Hence, the correct answer is option 'A': the rank of AtA is exactly 2.