In general, the eigenvalues of a diagonal matrix are given by the entries on the diagonal. verify this for 2 × 2 and 3 × 3 matrices.

Eigenvalues:

Eigenvalues of a matrix A are all the values of π for the following equation:

d = det (πI-A) = 0

For a 2x2 diagonal matrix:

Now,

d = det (πI-A)

π1-a1=0 and π2-a2=0

a1=π1 and a2=π2

Hence proved that the eigenvalues of a diagonal matrix are given by the entries on the diagonal.

For a 3x3 diagonal matrix:

Now,

d = det (πI-A)

π1-a1=0  ;  π2-a2=0  ;  π3-a3=0  

a1=π1   ;   a2=π2   ;   a3=π3

Hence proved that the eigenvalues of a diagonal matrix are given by the entries on the diagonal.