17 2.3 EXERCISESUnless otherwise specified, assume that all matrices in theseexercises are n xn. Determine which of the matrices in Exercises1-10 are invertible. Use as few calculations as possible. Justifyyour answers,12.1.3.5.7.[33]-65-38-13-209. [M]PARTE001notboos 930-44-6-110. [M] 7987-7 053bi bilo 2.0-9MOGENS09-1-33058-6 3-1 2-501mom274.til al-321of 6.8.-4 66-9-71 -503-361000302004119-1-7 -7119-5 101923-1 bis bud-4403 7 47475 90 200668110 in01100 75 397.des the642-8 igib to odman od539hoqon biss6 4 -9 1-5 to bolą ni beau5 2 118104(1) JAIn Exercises 11 and 12, the matrices are all n xn. Each part ofthe exercises is an implication of the form "If "statement 1",then "statement 2"." Mark an implication as True if the truth of"statement 2" always follows whenever "statement 1" happensto be true. An implication is False if there is an instance inwhich "statement 2" is false but "statement 1" is true. Justify eachanswer.11 a. If the equation Ax = 0 has only the trivial solution, thenA is row equivalent to the n x n identity matrix.b. If the columns of A span R", then the columns are linearlyindependent.c. If A is an n x n matrix, then the equation Ax = b has atleast one solution for each b in R".SAd. If the equation Ax = 0 has a nontrivial solution, then Ahas fewer than n pivot positions.e. If AT is not invertible, then A is not invertible.les asc. If the equation Ax = b has at least one solution for eachb in R", then the solution is unique for each b.2.3 Characterizations of Invertible Matrices 11713. An mxn upper triangular matrix is one whose entriesbelow the main diagonal are 0's (as in Exercise 8). Whenis a square upper triangular matrix invertible? Justify youranswer.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.d. If the linear transformation (x)→ Ax maps R" into R",then A has n pivot positions.e. If there is a b in R" such that the equation Ax=b isinconsistent, then the transformation x→ Ax is not one-to-one.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? (S) (1)17. If A is invertible, then the columns of A are linearlyindependent. Explain why.18. If C is 6 x 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?aocvdo bi19. 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 n x n 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 somenoiy in R", can the columns of G span R"? Why or why not?22. If the equation Hx = c is inconsistent for some e in R", whatcan you say about the equation Hx = 0? Why?23. If an n x n matrix K cannot be row reduced to I, what canyou say about the columns of K? Why?24. If L is n x n and the equation Lx = 0 has the trivial solution,do the columns of L span R"? Why?28.If there is an n x n matrix D such that AD = I, then thereis also an n x n matrix C such that CA = I.1809 BOITO 28. Shb. If the columns of A are linearly independent, then thecolumns of A span R".25. Verify the boxed statement preceding Example 1.26. Explain why the columns of A2 span R" whenever the(columns of A are linearly independent.29.27. Show that if AB is invertible, so is A. You cannot use Theorem6(b), because you cannot assume that A and B are invertible.[Hint: There is a matrix W such that ABW = I. Why?]Show that if AB is invertible, so is B.If A is an n x n matrix and the equation Ax=b has more thanone solution for some b, then the transformation x→ Ax isnot one-to-one. What else can you say about this transforma-tion? Justify your answer.

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Answered: 17. If A is invertible, then the…

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17. If A is invertible, then the columns of A¹ are linearly independent. Explain why.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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