Please answer all questions with detailed explanation thank you 4. (AB)-1 = A-B-15. Suppose AB = AC and A is invertible. Then B = C.6. If A is an invertible n x n matrix, then Ax = b is consistent for all b in R".7. If the columns of an n x n matrix A span R", then the columns of A are linearlyindependent.8. If the equation Ax = b has more than one solution for some b in R", then the columnsof A span R".9. If A is an n x n matrix and the equation Ax = 0 has a nontrivial solution, then A hasfewer than n pivots.10. Every square triangular matrix is invertible.11. Every line in R" is a subspace of R".12. Every line through the origin in R" is a subspace of R".13. The dimension of Nul(A) is the number of variables in the equation Ax = 0.14. The dimension of Col(A) is the number of pivot columns of A.15. Col(A) is the set of solutions to Ax = b.%3D

Answered: Image /qna-images/answer/bcdba3f4-e86e-42e3-b591-68557fffa14a.jpg

Answered: 4. (AB)-' = A-B-!

Math

Algebra

4. (AB)-' = A-B-!

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.2: Linear Independence, Basis, And Dimension

Problem 3AEXP

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Question

Please answer all questions with detailed explanation thank you

4. (AB)-1 = A-B-1
5. Suppose AB = AC and A is invertible. Then B = C.
6. If A is an invertible n x n matrix, then Ax = b is consistent for all b in R".
7. If the columns of an n x n matrix A span R", then the columns of A are linearly
independent.
8. If the equation Ax = b has more than one solution for some b in R", then the columns
of A span R".
9. If A is an n x n matrix and the equation Ax = 0 has a nontrivial solution, then A has
fewer than n pivots.
10. Every square triangular matrix is invertible.
11. Every line in R" is a subspace of R".
12. Every line through the origin in R" is a subspace of R".
13. The dimension of Nul(A) is the number of variables in the equation Ax = 0.
14. The dimension of Col(A) is the number of pivot columns of A.
15. Col(A) is the set of solutions to Ax = b.
%3D

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