Linear Algebra with Applications (2-Download)
5th Edition
ISBN: 9780321796974
Author: Otto Bretscher
Publisher: PEARSON
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Chapter 6.3, Problem 1E
Find the area of the parallelogram defined by
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1. Determine if
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polynomial identity.
2. Is (2x-y) (8x3+ y³) equivalent to
16x4-y4?
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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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- In a national park, the current population of an endangered species of bear is 80. Each year, the population decreases by 10%. How can you model the population of bears in the park? # of years # of bears 9 10 2 3 4 5 6 7 8 ° 1 Graph your data. Remember to title your graph. What scale should be used on the y-axis? What scale should be used on the x-axis? SMOKY 19 OUNTAINS NATIONAL Exponential Decay Equation y = a(1-r)* PARKarrow_forwardOn Feb. 8, this year, at 6am in the morning all UiB meteorology professors met to discuss a highly unfortunate and top-urgent crisis: Their most precious instrument, responsible for measuring the air temperature hour-by- hour, had failed - what if the Bergen public would find out? How would they plan their weekend without up-to-date air temperature readings? Silent devastation - and maybe a hint of panic, also - hung in the room. Apprentice Taylor, who - as always - was late to the meeting, sensed that this was his chance to shine! Could they fake the data? At least for some hours (until the measurements would work again)? He used to spend a lot of time online and thus knew the value of fake data, especially when it spread fast! He reminded the crying professors of a prehistoric project with the title "Love your derivatives as you love yourself!" - back then, they had installed top-modern technology that not only measured the air temperature itself, but also its 1st, 2nd, 3rd, 4th, and…arrow_forwardConsider a forest where the population of a particular plant species grows exponentially. In a real-world scenario, we often deal with systems where the analytical function describing the phenomenon is not available. In such cases, numerical methods come in handy. For the sake of this task, however, you are provided with an analytical function so that you can compare the results of the numerical methods to some ground truth. The population P(t) of the plants at time t (in years) is given by the equation: P(t) = 200 0.03 t You are tasked with estimating the rate of change of the plant population at t = 5 years using numerical differentiation methods. First, compute the value of P'(t) at t = 5 analytically. Then, estimate P'(t) at t = 5 years using the following numerical differentiation methods: ⚫ forward difference method (2nd-order accurate) 3 ⚫ backward difference method (2nd-order accurate) ⚫ central difference method (2nd-order accurate) Use h = 0.5 as the step size and round all…arrow_forward
- Nicole organized a new corporation. The corporation began business on April 1 of year 1. She made the following expenditures associated with getting the corporation started: Expense Date Amount Attorney fees for articles of incorporation February 10 $ 40,500 March 1-March 30 wages March 30 6,550 March 1-March 30 rent Stock issuance costs March 30 2,850 April 1-May 30 wages Note: Leave no answer blank. Enter zero if applicable. April 1 May 30 24,000 16,375 c. What amount can the corporation deduct as amortization expense for the organizational expenditures and for the start-up costs for year 1 [not including the amount determined in part (b)]? Note: Round intermediate calculations to 2 decimal places and final answer to the nearest whole dollar amount. Start-up costs amortized Organizational expenditures amortizedarrow_forwardLast Chance Mine (LCM) purchased a coal deposit for $2,918,300. It estimated it would extract 18,950 tons of coal from the deposit. LCM mined the coal and sold it, reporting gross receipts of $1.24 million, $13 million, and $11 million for years 1 through 3, respectively. During years 1-3, LCM reported net income (loss) from the coal deposit activity in the amount of ($11,400), $550,000, and $502,500, respectively. In years 1-3, LCM extracted 19,950 tons of coal as follows: (1) Tons of Coal 18,950 Depletion (2) Basis (2)(1) Rate $2,918,300 $154.00 Tons Extracted per Year Year 1 4,500 Year 2 8,850 Year 3 6,600 Note: Leave no answer blank. Enter zero if applicable. Enter your answers in dollars and not in millions of dollars. a. What is LCM's cost depletion for years 1, 2, and 3? Cost Depletion Year 1 Year 2 Year 3arrow_forwardConsider the following equation. log1/9' =6 Find the value of x. Round your answer to the nearest thousandth. x = ✓arrow_forward
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