Linear Algebra with Applications (2-Download)
5th Edition
ISBN: 9780321796974
Author: Otto Bretscher
Publisher: PEARSON
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Chapter 6, Problem 30E
There exist invertible
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Chapter 6 Solutions
Linear Algebra with Applications (2-Download)
Ch. 6.1 - Find the determinants of the matrices A in...Ch. 6.1 - Find the determinants of the matrices A in...Ch. 6.1 - Find the determinants of the matrices A in...Ch. 6.1 - Find the determinants of the matrices A in...Ch. 6.1 - Find the determinants of the matrices A in...Ch. 6.1 - Find the determinants of the matrices A in...Ch. 6.1 - Find the determinants of the matrices A in...Ch. 6.1 - Find the determinants of the matrices A in...Ch. 6.1 - Find the determinants of the matrices A in...Ch. 6.1 - Find the determinants of the matrices A in...
Ch. 6.1 - In Exercises 11 through 22, use the determinant to...Ch. 6.1 - In Exercises 11 through 22, use the determinant to...Ch. 6.1 - In Exercises 11 through 22, use the determinant to...Ch. 6.1 - In Exercises 11 through 22, use the determinant to...Ch. 6.1 - In Exercises 11 through 22, use the determinant to...Ch. 6.1 - In Exercises 11 through 22, use the determinant to...Ch. 6.1 - In Exercises 11 through 22, use the determinant to...Ch. 6.1 - In Exercises 11 through 22, use the determinant to...Ch. 6.1 - In Exercises 11 through 22, use the determinant to...Ch. 6.1 - In Exercises 11 through 22, use the determinant to...Ch. 6.1 - In Exercises 11 through 22, use the determinant to...Ch. 6.1 - In Exercises 11 through 22, use the determinant to...Ch. 6.1 - In Exercises 23 through 30, use the determinant to...Ch. 6.1 - In Exercises 23 through 30, use the determinant to...Ch. 6.1 - In Exercises 23 through 30, use the determinant to...Ch. 6.1 - In Exercises 23 through 30, use the determinant to...Ch. 6.1 - In Exercises 23 through 30, use the determinant to...Ch. 6.1 - In Exercises 23 through 30, use the determinant to...Ch. 6.1 - In Exercises 23 through 30, use the determinant to...Ch. 6.1 - In Exercises 23 through 30, use the determinant to...Ch. 6.1 - Find the determinants of the matrices A in...Ch. 6.1 - Find the determinants of the matrices A in...Ch. 6.1 - Find the determinants of the matrices A in...Ch. 6.1 - Find the determinants of the matrices A in...Ch. 6.1 - Find the determinants of the matrices A in...Ch. 6.1 - Find the determinants of the matrices A in...Ch. 6.1 - Find the determinants of the matrices A in...Ch. 6.1 - Find the determinants of the matrices A in...Ch. 6.1 - Find the determinants of the matrices A in...Ch. 6.1 - Find the determinants of the matrices A in...Ch. 6.1 - Find the determinants of the matrices A in...Ch. 6.1 - Find the determinants of the matrices A in...Ch. 6.1 - If A is an nn matrix, what is the relationship bet...Ch. 6.1 - If A is an nn matrix and k is an arbitrary...Ch. 6.1 - If A is a 22 matrix, what is the relationship...Ch. 6.1 - If A is an invertible 22 matrix, what is the...Ch. 6.1 - Find nonzero numbers a, b, c, d, e, f, g, h such...Ch. 6.1 - Find 22 matrices A, B, C, D such that...Ch. 6.1 - For two nonparallel vectors v and w in 3 ,...Ch. 6.1 - Prob. 50ECh. 6.1 - Explain why any pattern P in a matrix A, other...Ch. 6.1 - Consider two vectors v and w in 3 . Form the...Ch. 6.1 - Find the determinant of the (2n)(2n) matrix A=[0 I...Ch. 6.1 - Is the determinant of the matrix...Ch. 6.1 - Does the following matrix have an LU...Ch. 6.1 - Let Mn be the nn matrix with all 1‘s along “the...Ch. 6.1 - A square matrix is called a permutation matrix if...Ch. 6.1 - a. Find a noninvertible 22 matrix whose entries...Ch. 6.1 - Consider the function F(A)=F[vw]=vw from 22 to ,...Ch. 6.1 - Which of the following functions F of A=[abcd] are...Ch. 6.1 - Show that the function F[abcdefghj]=bfg is linear...Ch. 6.1 - In Exercises 62 through 64, consider a function D...Ch. 6.1 - In Exercises 62 through 64, consider a function D...Ch. 6.1 - In Exercises 62 through 64, consider a function D...Ch. 6.1 - Consider a function D from 33 to that is linearin...Ch. 6.1 - a. Let V be the linear space of all functions F...Ch. 6.2 - Use Gaussian elimination to find the determinant...Ch. 6.2 - Use Gaussian elimination to find the determinant...Ch. 6.2 - Use Gaussian elimination to find the determinant...Ch. 6.2 - Use Gaussian elimination to find the determinant...Ch. 6.2 - Use Gaussian elimination to find the determinant...Ch. 6.2 - Use Gaussian elimination to find the determinant...Ch. 6.2 - Use Gaussian elimination to find the determinant...Ch. 6.2 - Use Gaussian elimination to find the determinant...Ch. 6.2 - Use Gaussian elimination to find the determinant...Ch. 6.2 - Use Gaussian elimination to find the determinant...Ch. 6.2 - Consider a 44matrix A with rows v1,v2,v3,v4. If...Ch. 6.2 - Consider a 44matrix A with rows v1,v2,v3,v4. If...Ch. 6.2 - Consider a 44matrix A with rows v1,v2,v3,v4. If...Ch. 6.2 - Consider a 44matrix A with rows v1,v2,v3,v4. If...Ch. 6.2 - Consider a 44matrix A with rows v1,v2,v3,v4. If...Ch. 6.2 - Consider a 44matrix A with rows v1,v2,v3,v4. If...Ch. 6.2 - Find the determinants of the linear...Ch. 6.2 - Find the determinants of the linear...Ch. 6.2 - Find the determinants of the linear...Ch. 6.2 - Find the determinants of the linear...Ch. 6.2 - Find the determinants of the linear...Ch. 6.2 - Find the determinants of the linear...Ch. 6.2 - Find the determinants of the linear...Ch. 6.2 - Find the determinants of the linear...Ch. 6.2 - Find the determinants of the linear...Ch. 6.2 - Find the determinants of the linear...Ch. 6.2 - Find the determinants of the linear...Ch. 6.2 - Find the determinants of the linear...Ch. 6.2 - Let Pn be the nn matrix whose entries are all...Ch. 6.2 - Consider two distinct real numbers, a and b. We...Ch. 6.2 - Vandermonde determinants (introduced by...Ch. 6.2 - Use Exercise 31 to find...Ch. 6.2 - Prob. 33ECh. 6.2 - Prob. 34ECh. 6.2 - Consider two distinct points [a1a2] and [b1b2] in...Ch. 6.2 - Prob. 36ECh. 6.2 - Prob. 37ECh. 6.2 - Prob. 38ECh. 6.2 - If A is an invertible matrix, what can you say...Ch. 6.2 - If A is an orthogonal matrix, what are the...Ch. 6.2 - Consider a skew-symmetric nn matrix A. where nis...Ch. 6.2 - Consider an nm matrix A=QR ,where Q is an nm...Ch. 6.2 - Consider two vectors v and w in n . Form the...Ch. 6.2 - The cross product in n . Consider the vectors...Ch. 6.2 - Find the derivative of the function...Ch. 6.2 - Given some numbers a, b, c, d, e, and f such that...Ch. 6.2 - Is the function T[abcd]=ad+bc linear in the rows...Ch. 6.2 - Consider the linear transformation T(x)=det[ v 1...Ch. 6.2 - Give an example of a 33 matrix A with all nonzero...Ch. 6.2 - Find the determinant of the matrix...Ch. 6.2 - Find the determinant of the (2n)(2n) matrix A=[0 I...Ch. 6.2 - Consider a 22 matrix A=[abcd] with column vectors...Ch. 6.2 - Consider an invertible 22 matrix A with...Ch. 6.2 - Let A and B be 22 matrices with integer entries...Ch. 6.2 - For a fixed positive integer n, let D be a...Ch. 6.2 - Use the characterization of the determinant given...Ch. 6.2 - Consider a linear transformation T from m+n to m...Ch. 6.2 - Find the matrix M introduced in Exercise 57for the...Ch. 6.2 - If the equation detA=detB holds for two nn...Ch. 6.2 - Consider an nn matrix A. Show that swapping the...Ch. 6.2 - Consider nn matrices A. B. C, and D, where A...Ch. 6.2 - Consider nn matrices A, B , C, and D such that...Ch. 6.2 - Show that more than n!=123n multiplications are...Ch. 6.2 - Show that fewer than en! algebraic operations...Ch. 6.2 - Let Mn be the nn matrix with 1‘s on the main...Ch. 6.2 - Let Mn be the matrix with all 1‘s along the main...Ch. 6.2 - Consider a pattern P in an nn matrix, and choose...Ch. 6.2 - Using the terminology introduced in the proof of...Ch. 6.2 - Let G be the set of all integers x that can be...Ch. 6.2 - Throughout this exercise, consider the Fibonacci...Ch. 6.3 - Find the area of the parallelogram defined by [37]...Ch. 6.3 - Find the area of the triangle defined by [37] and...Ch. 6.3 - Prob. 3ECh. 6.3 - Consider the area A of the triangle with vertices...Ch. 6.3 - The tetrahedron defined by three vectors v1,v2,v3...Ch. 6.3 - What is the relationship between the volume of...Ch. 6.3 - Find the area of the following region:Ch. 6.3 - Demonstrate the equation |detA|=v1v2vn for a...Ch. 6.3 - If v1 and v2 are linearly independent vectors in 2...Ch. 6.3 - Consider an nn matrix A=[v1v2vn] .What is the...Ch. 6.3 - Consider a linear transformation T(x)=Ax from 2 to...Ch. 6.3 - Consider those 44 matrices whose entries are all...Ch. 6.3 - Prob. 13ECh. 6.3 - Find the 3-volume of the 3-parallelepiped defined...Ch. 6.3 - Demonstrate Theorem 6.3.6 for linearly dependent...Ch. 6.3 - True orfalse? If is a parallelogram in 3 and...Ch. 6.3 - (For some background on the cross product in n ,...Ch. 6.3 - If T(x)=Ax is an invertible linear transformation...Ch. 6.3 - A basis v1,v2,v3 of 3 is called positively...Ch. 6.3 - We say that a linear transformation T from 3 to 3...Ch. 6.3 - Arguing geometrically, determine whether the...Ch. 6.3 - Use Cramer’s rule to solve the systems in...Ch. 6.3 - Use Cramer’s rule to solve the systems in...Ch. 6.3 - Use Cramer’s rule to solve the systems in...Ch. 6.3 - Find the classical adjoint of the matrix...Ch. 6.3 - Consider an nn matrix A with integer entries such...Ch. 6.3 - Consider two positive numbers a and b. Solve the...Ch. 6.3 - In an economics text,10 we find the following...Ch. 6.3 - In an economics text11 we find the following...Ch. 6.3 - Find the classical adjointof A=[100230456] .Ch. 6.3 - Find the classical adjointof A=[111123166] .Ch. 6.3 - Find the classical adjointof A=[0001010000101000]...Ch. 6.3 - Find the classical adjointof A=[1000020000300004]...Ch. 6.3 - For an invertible nn matrix A, find the product...Ch. 6.3 - For an invertible nn matrix A, what is the...Ch. 6.3 - For an invertible nn matrix A, what is adj(adj A)?Ch. 6.3 - For an invertible nn matrix A, what is the...Ch. 6.3 - For two invertible nn matrices A and B, what is...Ch. 6.3 - If A and B are invertible nn matrices, and if A...Ch. 6.3 - For an invertible nn matrix A. consider the...Ch. 6.3 - Show that an nn matrix A has at least one...Ch. 6.3 - Even if an nn matrix A fails to be invertible, we...Ch. 6.3 - Show that A(adjA)=0(adjA)A for all noninvertible...Ch. 6.3 - If A isan nn matrixo frank n1 , what is the rank...Ch. 6.3 - Find all 22 matrices A such that adj(A)=AT .Ch. 6.3 - (For those who have studied multivariable...Ch. 6.3 - Consider the quadrilateral in the accompanying...Ch. 6.3 - What is the area of the largest ellipse you can...Ch. 6.3 - What are the lengths of the semi axes of the...Ch. 6 - If B is obtained be multiplying a column of A by...Ch. 6 - det(A10)=(detA)10 for all 1010 matrices A.Ch. 6 - The determinant of any diagonal nn matrix is the...Ch. 6 - If matrix B is obtained by swapping two rows of an...Ch. 6 - If A=[uvw] is any 33 matrix, then detA=u(vw) .Ch. 6 - det(4A)=4detA for all 44 matrices A.Ch. 6 - det(A+B)=4detA+detB for all 55 matrices A and B.Ch. 6 - The equation det(A)=detA holds for all 66...Ch. 6 - If all the entries of a 77 matrix A are 7, then...Ch. 6 - An 88 matrix fails to be invertible if (and only...Ch. 6 - Prob. 11ECh. 6 - Prob. 12ECh. 6 - Prob. 13ECh. 6 - Prob. 14ECh. 6 - Prob. 15ECh. 6 - Prob. 16ECh. 6 - Prob. 17ECh. 6 - Prob. 18ECh. 6 - Prob. 19ECh. 6 - Prob. 20ECh. 6 - If all the entries of a square matrix are 1 or 0,...Ch. 6 - If all the entries of a square matrix A are...Ch. 6 - If all the columns of a square matrix A are unit...Ch. 6 - If A is any noninvertible square matrix, then...Ch. 6 - If the determinant of a square matrix is 1 , then...Ch. 6 - If all the entries of an invertible matrix A are...Ch. 6 - There exists a 44 matrix A whose entries are all 1...Ch. 6 - If the determinant of a 22 matrix A is 4, then the...Ch. 6 - If A=[uvw] is a 33 matrix, then the formula...Ch. 6 - There exist invertible 22 matrices A and B such...Ch. 6 - There exist real invertible 33 matrices A and S...Ch. 6 - There exist real invertible 33 matrices A and S...Ch. 6 - If A is any symmetric matrix, then detA=1 or...Ch. 6 - If A is any skew-symmetric 44 matrix, then detA=0...Ch. 6 - If detA=detB for two nn matrices A and B, thenA...Ch. 6 - Suppose A is an nn matrix and B is obtained fromA...Ch. 6 - If an nn matrix A is invertible, then there must...Ch. 6 - If all the entries of matrices A and A1 are...Ch. 6 - If a square matrix A is invertible, then its...Ch. 6 - There exists a real 33 matrix A such that A2=I3 .Ch. 6 - If all the diagonal entries of an nn matrix A are...Ch. 6 - If all the diagonal entries of nn matrix A are...Ch. 6 - For every nonzero 22 matrix A there exists a 22...Ch. 6 - If A is a 44 matrix whose entries are all 1 or 1 ,...Ch. 6 - If A is an invertible nn matrix, then A must...Ch. 6 - There exists a real number k such that the matrix...Ch. 6 - If A and B are orthogonal nn matrices such that...
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- ma Classes Term. Spring 2025 Title Details Credit Hours CRN Schedule Type Grade Mode Level Date Status Message *MATHEMATICS FOR MANAGEME... MTH 245, 400 4 54835 Online Normal Grading Mode Ecampus Undergradu... 03/21/2025 Registered **Web Registered... *SOIL SCIENCE CSS 205, 400 0 52298 Online Normal Grading Mode Undergraduate 03/21/2025 Waitlisted Waitlist03/21/2025 PLANT PATHOLOGY BOT 451, 400 4 56960 Online Normal Grading Mode Undergraduate 03/21/2025 Registered **Web Registered... Records: 3 Schedule Schedule Detailsarrow_forwardHere is an augmented matrix for a system of equations (three equations and three variables). Let the variables used be x, y, and z: 1 2 4 6 0 1 -1 3 0 0 1 4 Note: that this matrix is already in row echelon form. Your goal is to use this row echelon form to revert back to the equations that this represents, and then to ultimately solve the system of equations by finding x, y and z. Input your answer as a coordinate point: (x,y,z) with no spaces.arrow_forward1 3 -4 In the following matrix perform the operation 2R1 + R2 → R2. -2 -1 6 After you have completed this, what numeric value is in the a22 position?arrow_forward
- 5 -2 0 1 6 12 Let A = 6 7 -1 and B = 1/2 3 -14 -2 0 4 4 4 0 Compute -3A+2B and call the resulting matrix R. If rij represent the individual entries in the matrix R, what numeric value is in 131? Input your answer as a numeric value only.arrow_forward1 -2 4 10 My goal is to put the matrix 5 -1 1 0 into row echelon form using Gaussian elimination. 3 -2 6 9 My next step is to manipulate this matrix using elementary row operations to get a 0 in the a21 position. Which of the following operations would be the appropriate elementary row operation to use to get a 0 in the a21 position? O (1/5)*R2 --> R2 ○ 2R1 + R2 --> R2 ○ 5R1+ R2 --> R2 O-5R1 + R2 --> R2arrow_forwardThe 2x2 linear system of equations -2x+4y = 8 and 4x-3y = 9 was put into the following -2 4 8 augmented matrix: 4 -3 9 This augmented matrix is then converted to row echelon form. Which of the following matrices is the appropriate row echelon form for the given augmented matrix? 0 Option 1: 1 11 -2 Option 2: 4 -3 9 Option 3: 10 ܂ -2 -4 5 25 1 -2 -4 Option 4: 0 1 5 1 -2 Option 5: 0 0 20 -4 5 ○ Option 1 is the appropriate row echelon form. ○ Option 2 is the appropriate row echelon form. ○ Option 3 is the appropriate row echelon form. ○ Option 4 is the appropriate row echelon form. ○ Option 5 is the appropriate row echelon form.arrow_forward
- Let matrix A have order (dimension) 2x4 and let matrix B have order (dimension) 4x4. What results when you compute A+B? The resulting matrix will have dimensions of 2x4. ○ The resulting matrix will be a single number (scalar). The resulting matrix will have dimensions of 4x4. A+B is undefined since matrix A and B do not have the same dimensions.arrow_forwardIf -1 "[a446]-[254] 4b = -1 , find the values of a and b. ○ There is no solution for a and b. ○ There are infinite solutions for a and b. O a=3, b=3 O a=1, b=2 O a=2, b=1 O a=2, b=2arrow_forwardA student puts a 3x3 system of linear equations is into an augmented matrix. The student then correctly puts the augmented matrix into row echelon form (REF), which yields the following resultant matrix: -2 3 -0.5 10 0 0 0 -2 0 1 -4 Which of the following conclusions is mathematically supported by the work shown about system of linear equations? The 3x3 system of linear equations has no solution. ○ The 3x3 system of linear equations has infinite solutions. The 3x3 system of linear equations has one unique solution.arrow_forward
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