17 14. An mxn lower triangular matrix is one whose entriesabove the main diagonal are 0's (as in Exercise 3). Whenis a square lower triangular matrix invertible? Justify youranswer.15. Can a square matrix with two identical columns be invert-ible? Why or why not?16. Is it possible for a 5 x 5 matrix to be invertible when itscolumns do not span R5? Why or why not?17. If A is invertible, then the columns of A-¹ are linearlyindependent. Explain why.18. If C is 6 × 6 and the equation Cx = v is consistent for everyv in R6, is it possible that for some v, the equation Cx = vhas more than one solution? Why or why not?gqu?19. If the columns of a 7 x 7 matrix D are linearly independent,what can you say about solutions of Dx = b? Why?20. If nxn matrices E and F have the property that EF = 1,then E and F commute. Explain why.21. If the equation Gx = y has more than one solution for somey in R", can the columns of G span R"? Why or why not?2. If the equation Hx = c is inconsistent for some c in R", whatcan you say about the equation Hx = 0? Why?3. If an n x n matrix K cannot be row reduced to In, what canyou say about the columns of K? Why?4. If L is n x n and the equation Lxdo the columns of L span R"? Why?-0 has the trivial solution,

Answered: Image /qna-images/answer/02fd28c6-44a9-42dd-b1e8-97776078284b.jpg

An mxn lower triangular matrix is one whose entries above the main diagonal are 0's (as in Exercise 3). When is a square lower triangular matrix invertible? Justify your answer. 15. Can a square matrix with two identical columns be invert- ible? Why or why not? 16. Is it possible for a 5 x 5 matrix to be invertible when its columns do not span R5? Why or why not? 17. If A is invertible, then the columns of A are linearly independent. Explain why. 18. If C is 6 x 6 and the equation Cx = v is consistent for every v in R6, is it possible that for some v, the equation Cx = v has more than one solution? Why or why not? 209002 TE 19. If the columns of a 7 x 7 matrix D are linearly independent, what can you say about solutions of Dx = b? Why? 20. If nxn matrices E and F have the property that EF = 1, then E and F commute. Explain why. 21. If the equation Gx = y has more than one solution for some y in R", can the columns of G span R"? Why or why not? 2. If the equation Hx = c is inconsistent for some c in R", what can you say about the equation Hx = 0? Why? 3. If an n x n matrix K cannot be row reduced to In, what can you say about the columns of K? Why? and the equation Ix-0 hao th TE

An mxn lower triangular matrix is one whose entries above the main diagonal are 0's (as in Exercise 3). When is a square lower triangular matrix invertible? Justify your answer. 15. Can a square matrix with two identical columns be invert- ible? Why or why not? 16. Is it possible for a 5 x 5 matrix to be invertible when its columns do not span R5? Why or why not? 17. If A is invertible, then the columns of A are linearly independent. Explain why. 18. If C is 6 x 6 and the equation Cx = v is consistent for every v in R6, is it possible that for some v, the equation Cx = v has more than one solution? Why or why not? 209002 TE 19. If the columns of a 7 x 7 matrix D are linearly independent, what can you say about solutions of Dx = b? Why? 20. If nxn matrices E and F have the property that EF = 1, then E and F commute. Explain why. 21. If the equation Gx = y has more than one solution for some y in R", can the columns of G span R"? Why or why not? 2. If the equation Hx = c is inconsistent for some c in R", what can you say about the equation Hx = 0? Why? 3. If an n x n matrix K cannot be row reduced to In, what can you say about the columns of K? Why? and the equation Ix-0 hao th TE

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.7: Applications

Problem 41EQ

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