522 CHAPTER 27. DETERMINANTS

Show that for W (x) =W (y1, · · · ,yn)(x) to save space,

W ′ (x) = det

y1 (x) · · · yn (x)y′1 (x) · · · y′n (x)

......

y(n)1 (x) · · · y(n)n (x)

 .

Now use the differential equation, Ly = 0 which is satisfied by each of these func-tions, yi and properties of determinants presented above to verify the differentialequation W ′ + an−1 (x)W = 0. Give an explicit solution of this linear differentialequation, Abel’s formula, and use your answer to verify that the Wronskian of thesesolutions to the equation, Ly = 0 either vanishes identically on (a,b) or never. Hint:To solve the differential equation, let A′ (x) = an−1 (x) and multiply both sides of thedifferential equation by eA(x) and then argue the left side is the derivative of some-thing.

47. Find the following determinants and the inverses of the given matrices. You mightuse MATLAB to do this with no trouble.

(a) det

 2 2+2i 3−3i2−2i 5 1−7i3+3i 1+7i 16

 (b) det

 10 2+6i 8−6i2−6i 9 1−7i8+6i 1+7i 17

48. Find the eigenvalues and eigenvectors of the following matrices. List the eigenvalues

according to multiplicity as a root of the characteristic polyinomial.

(a)

 4 7 5−2 −4 −41 3 4



(b)

 1 1 20 0 −20 1 3



(c)

 −3 −7 −24 8 2−2 −3 1



(d)

 4 6 3−2 −3 −21 2 2

49. The eigenspace for an eigenvalue λ is defined to be the span of all eigenvectors. If

the dimension of the eigenspace for each λ equals the multiplicity of the eigenvalueas a root of the characteristic polynomial, then the matrix is said to be nondefective.If, for any eigenvalue, the dimension of the eigenspace called geometric multiplicity