Step 1: Use the trace and determinant properties:
For a 3 by 3 matrix, the sum of the eigenvalues equals the trace of the matrix, and the product of the eigenvalues equals the determinant of the matrix. The trace of the given matrix is $8 + 5 + 7 = 20$, and since the matrix is diagonal, its determinant is the product of the diagonal entries, $8 \times 5 \times 7 = 280$.
Step 2: Verify with each candidate option:
For option A, $\lambda = 1, 1, 1$ gives a sum of 3 and a product of 1, neither of which matches. For option C, $\lambda = 0, 0, 0$ gives a sum of 0 and a product of 0, which also does not match. For option D, $\lambda = 40, 1, 35$ gives a sum of 76 and a product of 1400, which again does not match.
Step 3: Confirm option B directly:
For option B, $\lambda = 8, 5, 7$ gives a sum of $8 + 5 + 7 = 20$, which equals the trace, and a product of $8 \times 5 \times 7 = 280$, which equals the determinant. Both conditions are satisfied simultaneously only by this set of values.
Step 4: Confirm through eigenvectors:
For a diagonal matrix, each standard basis vector is itself an eigenvector. For instance, $e_1 = (1, 0, 0)$ satisfies $Ae_1 = 8e_1$, so 8 is an eigenvalue. Similarly $e_2$ gives eigenvalue 5 and $e_3$ gives eigenvalue 7, confirming the result independently.
Final Answer:
\[ \boxed{\lambda = 8, 5, 7} \]