88 CHAPTER 3. DETERMINANTS

8. Let A and B be two n× n matrices. A ∼ B (A is similar to B) means there exists aninvertible matrix S such that A = S−1BS. Show that if A ∼ B, then B ∼ A. Showalso that A ∼ A and that if A ∼ B and B ∼ C, then A ∼ C.

9. In the context of Problem 8 show that if A ∼ B, then det (A) = det (B) .

10. Let A be an n× n matrix and let x be a nonzero vector such that Ax = λx for somescalar, λ. When this occurs, the vector, x is called an eigenvector and the scalar, λis called an eigenvalue. It turns out that not every number is an eigenvalue. Onlycertain ones are. Why? Hint: Show that if Ax = λx, then (λI −A)x = 0. Explainwhy this shows that (λI −A) is not one to one and not onto. Now use Theorem 3.1.15to argue det (λI −A) = 0. What sort of equation is this? How many solutions does ithave?

11. Suppose det (λI −A) = 0. Show using Theorem 3.1.15 there exists x ̸= 0 such that(λI −A)x = 0.

12. 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?

13. Use the formula for the inverse in terms of the cofactor matrix to find the inverse ofthe matrix

A =

 et 0 0

0 et cos t et sin t

0 et cos t− et sin t et cos t+ et sin t

 .

14. Let A be an r×r matrix and let B be an m×m matrix such that r+m = n. Considerthe following n× n block matrix

C =

(A 0

D B

).