Linear Algebra, Theory And Applications, 2012a

82 DETERMINANTS
(b)
(c)
⎛
⎝
⎛
⎜
⎝
4 3 2
1 7 8
3 −9 3
1 2 3 2
1 3 2 3
4 1 5 0
1 2 1 2
⎞
⎠(The answer is 375.)
⎞
⎟
⎠ ,(Theansweris−2.)
2. If A −1 exist, what is the relationship between det (A) and det ( A −1) . Explain your
answer.
3. Let A be an n × n matrix where n is odd. Suppose also that A is skew symmetric.
This means A T = −A. Show that det(A) =0.
4. Is it true that det (A + B) =det(A)+det(B)? If this is so, explain why it is so and
if it is not so, give a counter example.
5. Let A be an r ×r matrix and suppose there are r −1 rows (columns) such that all rows
(columns) are linear combinations of these r − 1 rows (columns). Show det (A) =0.
6. Show det (aA) =a n det (A) where here A is an n × n matrix and a is a scalar.
7. Suppose A is an upper triangular matrix. Show that A −1 exists if and only if all
elements of the main diagonal are non zero. Is it true that A −1 will also be upper
triangular? Explain. Is everything the same for lower triangular matrices?
8. Let A and B be two n × n matrices. A ∼ B (A is similar to B) means there exists an
invertible matrix S such that A = S −1 BS. Show that if A ∼ B, then B ∼ A. Show
also 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 some
scalar, λ. When this occurs, the vector, x is called an eigenvector and the scalar, λ
is called an eigenvalue. It turns out that not every number is an eigenvalue. Only
certain ones are. Why? Hint: Show that if Ax = λx, then (λI − A) x = 0. Explain
why this shows that (λI − A) is not one to one and not onto. Now use Theorem 3.1.15
to argue det (λI − A) =0. What sort of equation is this? How many solutions does it
have?
11. Suppose det (λI − A) =0. Show using Theorem 3.1.15 there exists x ≠ 0 such that
(λI − A) x = 0.
( )
a (t) b (t)
12. Let F (t) =det
. Verify
c (t) d (t)
(
F ′ a
(t) =det
′ (t) b ′ (t)
c (t) d (t)
) (
a (t) b (t)
+det
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)
⎞
⎠ .