126 CHAPTER 6. DETERMINANTS

42. Here is a matrix, et e−t cos t e−t sin tet −e−t cos t− e−t sin t −e−t sin t + e−t cos tet 2e−t sin t −2e−t cos t

Does there exist a value of t for which this matrix fails to have an inverse? Explain.

43. Suppose A is an upper triangular matrix. Show that A−1 exists if and only if allelements of the main diagonal are non zero. Is it true that A−1 will also be uppertriangular? Explain. Is everything the same for lower triangular matrices?

44. If A,B, and C are each n× n matrices and ABC is invertible, why are each of A,B,and C invertible.

45. Let F (t) = det

(a(t) b(t)c(t) d (t)

). Verify

F ′ (t) = det

(a′ (t) b′ (t)c(t) d (t)

)+det

(a(t) b(t)c′ (t) d′ (t)

).

Now suppose

F (t) = det

 a(t) b(t) c(t)d (t) e(t) f (t)g(t) h(t) i(t)

 .

Use Laplace expansion and the first part to verify F ′ (t) =

det

 a′ (t) b′ (t) c′ (t)d (t) e(t) f (t)g(t) h(t) i(t)

+det

 a(t) b(t) c(t)d′ (t) e′ (t) f ′ (t)g(t) h(t) i(t)

+det

 a(t) b(t) c(t)d (t) e(t) f (t)g′ (t) h′ (t) i′ (t)

 .

Conjecture a general result valid for n× n matrices and explain why it will be true.Can a similar thing be done with the columns?

46. Let Ly = y(n)+ an−1 (x)y(n−1)+ · · ·+ a1 (x)y′+ a0 (x)y where the ai are given con-tinuous functions defined on a closed interval, (a,b) and y is some function whichhas n derivatives so it makes sense to write Ly. Suppose Lyk = 0 for k = 1,2, · · · ,n.The Wronskian of these functions, yi is defined as

W (y1, · · · ,yn)(x)≡ det

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

......

y(n−1)1 (x) · · · y(n−1)

n (x)

