Page 205 - Discrete Mathematics and Its Applications

184 2 / Basic Structures: Sets, Functions, Sequences, Sums, and Matrices

4. Find the product AB, where 15. Let
⎡ ⎤ ⎡ ⎤
1 0 1 0 1 −1

a) A = ⎣ 0 −1 −1 ⎦ , B = ⎣ 1 −1 0 ⎦ . A = 11 .
−1 1 0 −1 0 1 0 1
⎡ ⎤ ⎡ ⎤
1 −3 0 1 −1 2 3 n
b) A = ⎣ 1 2 2 ⎦ , B = ⎣ −1 0 3 −1 ⎦ . Find a formula for A , whenever n is a positive integer.
t t
2 1 −1 −3 −2 0 2 16. Show that (A ) = A.
⎡ ⎤ 17. Let A and B be two n × n matrices. Show that
0 −1 4 −12 30 t t t
c) A = ⎣ 7 2 ⎦ , B = −2 0 3 4 1 . a) (A + B) = A + B .
t
t
t
−4 −3 b) (AB) = B A .
5. Find a matrix A such that If A and B are n × n matrices with AB = BA = I n , then B
is called the inverse of A (this terminology is appropriate be-

2 3 30 cause such a matrix B is unique) and A is said to be invertible.
A = . −1
14 1 2 The notation B = A denotes that B is the inverse of A.
18. Show that
[Hint: Finding A requires that you solve systems of linear
⎡ ⎤
equations.] 2 3 −1
⎣ 1 2
6. Find a matrix A such that 1 ⎦
−1 −1 3
⎡ ⎤ ⎡ ⎤
1 3 2 7 1 3
⎣ 2 1 1 ⎦ A = ⎣ 1 0 3 ⎦ . is the inverse of
4 0 3 −1 −37 ⎡ ⎤
7 −8 5
⎣ −4 5 −3 ⎦ .
7. Let A be an m × n matrix and let 0 be the m × n matrix 1 −1 1
that has all entries equal to zero. Show that A = 0 + A =
A + 0. 19. Let A be the 2 × 2 matrix
8. Show that matrix addition is commutative; that is,

show that if A and B are both m × n matrices, then a b
A = .
A + B = B + A. c d
9. Show that matrix addition is associative; that is, show
that if A, B, and C are all m × n matrices, then Show that if ad − bc = 0, then
A + (B + C) = (A + B) + C.
d −b
⎡ ⎤
10. Let A bea3 × 4 matrix, B bea4 × 5 matrix, and C be a ⎢ ad − bc
4 × 4 matrix. Determine which of the following products A −1 = ⎢ ad − bc ⎥ .
⎥
⎣ −c a ⎦
are defined and find the size of those that are defined.
a) AB b) BA c) AC ad − bc ad − bc
d) CA e) BC f) CB
20. Let
11. What do we know about the sizes of the matrices A and
B if both of the products AB and BA are defined? −12
A = .
12. In this exercise we show that matrix multiplication is dis- 1 3
tributive over matrix addition.
a) Suppose that A and B are m × k matrices and that C a) Find A −1 .[Hint: Use Exercise 19.]
3
is a k × n matrix. Show that (A + B)C = AC + BC. b) Find A .
) .
b) SupposethatCisanm × k matrixandthatAandBare c) Find (A −1 3
)
k × n matrices. Show that C(A + B) = CA + CB. d) Use your answers to (b) and (c) to show that (A −1 3
3
13. In this exercise we show that matrix multiplication is is the inverse of A .
n −1
associative. Suppose that A is an m × p matrix, B is 21. Let A be an invertible matrix. Show that (A ) =
) whenever n is a positive integer.
a p × k matrix, and C is a k × n matrix. Show that (A −1 n
A(BC) = (AB)C. 22. Let A be a matrix. Show that the matrix AA is symmet-
t
14. The n × n matrix A =[a ij ] is called a diagonal matrix if ric. [Hint: Show that this matrix equals its transpose with
a ij = 0 when i = j. Show that the product of two n × n the help of Exercise 17b.]
diagonal matrices is again a diagonal matrix. Give a sim- 23. Suppose that A is an n × n matrix where n is a positive
t
ple rule for determining this product. integer. Show that A + A is symmetric.