Explanation:
To find the third eigenvalue of a third-order matrix A, we can use the fact that the sum of the eigenvalues of a matrix is equal to the trace of the matrix, and the product of the eigenvalues is equal to the determinant of the matrix.

Method:
Using these facts, we can write the following equations:
- The sum of the eigenvalues = trace(A) = a11 + a22 + a33
- The product of the eigenvalues = det(A) = (2)(1)(λ3) = 2λ3

We are given the two eigenvalues of A as 2 and 1 + 2i. We can use the fact that A is a real matrix to conclude that the third eigenvalue must be 1 - 2i (complex conjugate of 1 + 2i).

Solution:
The sum of the eigenvalues of A = 2 + (1 + 2i) + (1 - 2i) = 4
The product of the eigenvalues of A = 2(1 + 2i)(1 - 2i) = 10

Now, we can use the fact that the sum of the eigenvalues is equal to the trace of the matrix to find the value of a33:
a11 + a22 + a33 = 4
a33 = 4 - a11 - a22

We can also use the fact that the product of the eigenvalues is equal to the determinant of the matrix to find the value of a33:
det(A) = (2)(1)(λ3) = 10
a11a22a33 + a12a23a31 + a21a32a13 - a31a22a13 - a32a23a11 - a21a12a33 = 10

Using these two equations, we can solve for a33:
a33 = 3 - a11 - a22 = 5/3

Therefore, the third eigenvalue of A is 1 - 2i.