Is λ=2\lambda = 2λ=2 an eigenvalue of A=(1203)A = \begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}A=(1023)?
Yes, because 222 lies between the diagonal entries.
No, because det(A−2I)≠0\det(A - 2I) \neq 0det(A−2I)=0.
Yes, always for diagonal matrices.
Cannot determine without computing eigenvectors.