Find the eigenvalues of the following matrix.
Recall that the eigenvalues of a matrix are the roots of the polynomial . Now in this case, we have
Using the Cofactor Expansion Formula for the determinant along the 4th row, (Theorem 29 on page 439 of D&F, with ), we have that . Since , this simplifies to
Note that both of these minor matrices are diagonal, and recall that the determinant of an upper- or lower-triangular matrix is the product of the diagonal entries. (To see this, consider the Leibniz formula for the determinant: . If is upper (lower) triangular, then for every , if moves at least one element in , then it must move some element to a strictly larger (smaller) one, in which case vanishes since some factor is zero. What remains is merely .)
So we have . Clearly the roots of this polynomial, hence the eigenvalues of , are and .