faster than computing both A and AB. 13. Suppose AB = AC, where B and C are n x p matrices and A is invertible. Show that B = C. Is this true, in general, when A is not invertible?
faster than computing both A and AB. 13. Suppose AB = AC, where B and C are n x p matrices and A is invertible. Show that B = C. Is this true, in general, when A is not invertible?
Operations Research : Applications and Algorithms
4th Edition
ISBN:9780534380588
Author:Wayne L. Winston
Publisher:Wayne L. Winston
Chapter2: Basic Linear Algebra
Section2.1: Matrices And Vectors
Problem 5P
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Question
13
![1
112 CHAPTER 2 Matrix Algebra
If [A B]
[IX], then X = A-¹ B.
If A is larger than 2 x 2, then row reduction of [A B] is much
faster than computing both A and A-¹ B.
~II~
13. Suppose AB = AC, where B and C are n x p matrices and A
is invertible. Show that B = C. Is this true, in general, when
A is not invertible?
14. Suppose (B-C)D = 0, where B and C are m x n matrices
and D is invertible. Show that B = C.
15. Suppose A, B, and C are invertible n x n matrices. Show that
ABC is also invertible by producing a matrix D such that
(ABC) D = I and D (ABC) = I.
16. Suppose A and B are n x n, B is invertible, and AB is invert-
ible. Show that A is invertible. [Hint: Let C = AB, and solve
this equation for A.]
17. Solve the equation AB = BC for A, assuming that A, B, and
C are square and B is invertible.
18. Suppose P is invertible and A = PBP-¹. Solve for B in
terms of A.
19. If A, B, and C are n x n invertible matrices, does the equation
C-¹(A + X)B-1 = I have a solution, X? If so, find it.
20. Suppose A, B, and X are nxn matrices with A, X, and
A- AX invertible, and suppose
(A - AX)¹ = X-¹ B
(3)
a. Explain why B is invertible.
b. Solve (3) for X. If you need to invert a matrix, explain
why that matrix is invertible.
21. Explain why the columns of an n x n matrix A are linearly
independent when A is invertible.
22. Explain why the columns of an n x n matrix A span R" when
A is invertible. [Hint: Review Theorem 4 in Section 1.4.]
23. Suppose A is n x n and the equation Ax = 0 has only the
trivial solution. Explain why A has n pivot column
row equivalent to
3 x=
more
27.
28.
Find
Use
29.
31.
33.
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Transcribed Image Text:1
112 CHAPTER 2 Matrix Algebra
If [A B]
[IX], then X = A-¹ B.
If A is larger than 2 x 2, then row reduction of [A B] is much
faster than computing both A and A-¹ B.
~II~
13. Suppose AB = AC, where B and C are n x p matrices and A
is invertible. Show that B = C. Is this true, in general, when
A is not invertible?
14. Suppose (B-C)D = 0, where B and C are m x n matrices
and D is invertible. Show that B = C.
15. Suppose A, B, and C are invertible n x n matrices. Show that
ABC is also invertible by producing a matrix D such that
(ABC) D = I and D (ABC) = I.
16. Suppose A and B are n x n, B is invertible, and AB is invert-
ible. Show that A is invertible. [Hint: Let C = AB, and solve
this equation for A.]
17. Solve the equation AB = BC for A, assuming that A, B, and
C are square and B is invertible.
18. Suppose P is invertible and A = PBP-¹. Solve for B in
terms of A.
19. If A, B, and C are n x n invertible matrices, does the equation
C-¹(A + X)B-1 = I have a solution, X? If so, find it.
20. Suppose A, B, and X are nxn matrices with A, X, and
A- AX invertible, and suppose
(A - AX)¹ = X-¹ B
(3)
a. Explain why B is invertible.
b. Solve (3) for X. If you need to invert a matrix, explain
why that matrix is invertible.
21. Explain why the columns of an n x n matrix A are linearly
independent when A is invertible.
22. Explain why the columns of an n x n matrix A span R" when
A is invertible. [Hint: Review Theorem 4 in Section 1.4.]
23. Suppose A is n x n and the equation Ax = 0 has only the
trivial solution. Explain why A has n pivot column
row equivalent to
3 x=
more
27.
28.
Find
Use
29.
31.
33.
34.
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