Linear Algebra Questions and Answers

Find a PTLU factorization of the given matrix A Enter each matrix in the form row 1 row 2 where each row is a comma separated I A PT L U 014

Find an LU factorization of the given matrix 3 3 1 95 4 3 4 21 4 L I X 000 3 0 0 1 1 4 3 4 0 0 1 1 7

Find a basis B for the span of the given vectors 1 HOA 7 0 B 5 6 1 000 7 1

Consider the following 1 1 0 2 1 1 10 Find Av Av A 000 9 1 V 19 1 2 Complete the following statement Since Av 0 v Select in null A

Find an LU factorization of the given matrix 3 3 1 B3 95 4 3 4 21 4 L U 100 000 100 000 000 00 188

4 0 BOD 4 0 4 4 4 4 0 What is the rank of the matrix with the given vectors as its rows rank Do the given vectors form a basis for R Yes they form a basis for R No they do not form a basis for R

Give the rank and nullity of the matrix below A rank A nullity A 2 4 0 2 1 1 2 1 28 1 2 1 49

Consider the following Find Av Av 1 0 1 A 181 1 v Q V 9 1 Complete the following statement Since Av 0 V Select in nulla

For an Invertible matrix with an LU factorization A LU both L and U will be invertible and A U L1 Find L1 U 1 and A 1 for the given matri CHAB U 1 A 1 A

3 rea A of the triangle whose sides have the given lengths Round your answer to the 14 b 109 C 110 Help DETAILS ts Read It DETAILS area A of the triangle whose sides have the given lengths a 6 b 10 Help Read It C 8 Watch It

is a system of equations y 3x 1 y 3x 3 Graph the system Then write its solution Note that you can also answer No solution or Infinitely many solutio X

Judy desires to increase both her protein consumption and caloric intake She desires to have at least 60 more grams of protein each day and no more than an additional 350 calories daily An ounce of parmesan cheese has 6 grams of protein and 20 calories An ounce of cheddar cheese has 10 grams of protein and 50 calories Write a system of 2 inequalities to model this situation Use z for the ounces of parmesan cheese and y for the ounces of cheddar cheese NOTE You cannot have less than none First Inequality Second Inequality 13 12 11 10 Solve your system of inequalities by graphing 15 14 69 8 7 6 5 32 Select an answer 350 2 Select an answer 60

Caitlyn sells her drawings at the county fair She wants to sell at least 85 drawings and has portraits and landscapes She sells the portraits for 10 each and the landscapes for 16 each She needs to sell at least 640 worth of drawings in order to earn a profit Write a system of 2 inequalities to model this situation Use z for the number of portraits and y for the number of landscapes First Inequality Second Inequality 110 100 Solve your system of inequalities by graphing 90 80 70 60 50 40 30 20 10 190 Select an answer 85 Select an answer 640 1 10 20 30 40 50 60 70 80 90 100110

ces TB MC Qu 03 55 Static Which one of the followi Which one of the following would not be included in a closing er Multiple Choice A debit to Unearned Revenue A credit to Dividends

box provided Use the tools and your keyboa 16g 5 40g 4 24g 3

Select the correct choice below and if necessary fill in the answer box to complete your choice A The solution set in interval notation is Simplify your answer Use integers or fractions for any numbers in the expression Type your answer in interval notation O The solution set is Graph the solution set on a number line Choose the correct graph below OA OE The solution set is OB OF

A STORY OF UNITS 4 Mr Ramirez receives 4 gets of books Each set has 16 fiction books and 14 nonfiction books He puts 97 books in his library and donates the rest How many books does he donate 16 14 12124 Lesson 21 Homework 33 1224456 97 137 5 Cella sells calendars for a fundraiser Each calendar costs 9 She selle kalendars to her family members and 14 calendars to the people in her neighborhood Her goal is to earn 300 Does Cella reach her goal Explain your answer 16px 9 144 The video store sells science and history movies for 5 each How much money does the video store make if it sells 33 science movies and 57 history movjes

Question 5 Submit Question Find the magnitude of the projection of 3 6 onto the vector 3 1

Question 8 Let a 7i 53 and 6 4i 33 Find a b

If A 1 15 then A is invertible if ad bc 0 in which case If ad bc 0 then A is not invertible Find the Inverse of the given matrix if it exists using the theorem above If the inverse is not possible enter DNE in any single blan 0 08 1 A ad bc 6 8 1 5 8 6 0 5 6 8 1 46 0 25 11

Example 3 25 uses the inverse of the coefficient matrix to solve a linear system This method can be used to solve the following system X1 X 1 9X X 9 State the coefficient matrix A Order the columns from x to x Do not perform any row operations A Find the inverse of the coefficient matrix A Give b such that Ax b b Solve the given system using the method of Example 3 25

HA 8 If then A is invertible if ad bc 0 in which case A 1 E 1 ad bc If ad bc 0 then A is not invertible Find the Inverse of the given matrix if it exists using the theorem above If the inverse is not possible enter DNE in any single bla d b 1 5 8 6 0 5 6 8

Solve the given matrix equation for X Simplify your answers as much as possible In the words of Albert Einstein Everything should be made as simple as possible but not simpler Assume that all matrices are invertible AXB BA O X BA OX A BAB 1 OX I O X A BA B 1 OX A 8 1

If A then A is invertible if ad bc 0 in which case 4 ad Dc A 1 bc If ad bc 0 then A is not invertible Find the Inverse of the given matrix if it exists using the theorem above If this is not possible enter DNE In any single bla

a Give an example to show that if A and B are symmetric n x n matrices then AB need not be symmetric 02 OA 2 3 B 1 4 which is not symmetric OA 3 2 t B 10 23 OA 20 13 20 OA A OA B B but AB but AB but AB but AB 02 2 81 0 4 81 0 01 1 but AB which is not symmetric which is not symmetric which is not symmetric which is not symmetric b Prove that If A and B are symmetric n x n matrices then AB is symmetric if and only if AB BA O Suppose A and B are symmetric Then AB AB and BA BA Thus AB is symmetric If and only if AB AB BA E Suppose A and B are symmetric Then AB BTAT BA Thus AB is symmetric if and only if AB AB BA Suppose A and B are symmetric Then AB ATBT AB Thus AB Is symmetric If and only if BA AB AB Suppose A and B are symmetric Then AB BAT BA Thus AB is symmetric if and only if AB AB BA Suppose A and B are symmetric Then AB ATBT AB Thus AB Is symmetric if and only if 8A AB AB

a b If B c 0 O a b a d O c d b c find conditions on a b c and d such that AB BA Select all necessary conditions A 14 1 1 1

DETAILS an are that subtends a central angle of 3 rad in a circle DETAILS Wwich It PREVIOUS ANSWERS an arc that subtends a central angle of 45 in a circle m

Use the function to on the coordinate f x 2x 1 REMEME

Solve the following system of equations 8x 3y 15 7x 5y 6 X 0 010 X

A survey found that people keep their televisions an average of 5 9 years The standard deviation is 0 72 years Find the percentage of people that owne their TV for the given amount of time Assume the random variable is normally distributed Refer to the table of values Area Under the Standard Normal Distribution as needed Part 0 3 Part 1 of 3 a More than 6 62 years The percentage of people who owned his or her TV for more than 6 62 years is X

ifx21 a Graph the function with a graphing utility T N 2 0 2 N

ing equal the equation of the circle below 5 y 5 0 5 O 5 X O x y 12x 45 0 O x y 12x 27 0 Tepie 10

work CIT04 Suppose that there are three plots of mountain resort land available for sale in Breckenridge and s one plot Assume that all of the plots are basically indistinguishable and that the minimum selling p each potential buyer s willingness and ability to purchase a plot of land Person Andrew Beth Lorenzo Neha Sam Teresa Willingness and Ability to Purchase Dollars 720 000 700 000 900 000 850 000 790 000 740 000 Which of these people will purchase one of the three mountain resort plots Check all that apply Andrew

If 8 workers complete a certain task in 3 days how many workers are needed to finish the same task in 4 day workers

Homework Ch 04 The following graph shows the monthly demand and supply curves in the market for combs Use the graph input tool to help you answer the following questions You will not be graded Note Once you enter a value in a white field the graph and any corresponding amounts in CE Dollars per comb 100 90 80 70 60 50 40 Supply Demand Graph Input Tool Market for Combs Price Dollars per comb Quantity Demanded Combs

Write B as a linear combination of the other matrices if possible If not enter DNE in all blanks 0 24 A3 B 32 4 1 42 i 1 A B 0 0 4 O SHOH 24 5 1 0 1 1

and Ab Use the matrix column representation of the product to write each column of AB as a linear combination of the columns of A Ab 2 11 Ab 3 1 2 2 A 3 1 1 20 1 B AB 3 2 30 1 1 1 1 2 4 6 X HO El Give the final product AB DEL 1 12 1 6 5 BOH i 2 i G 2 6 5 1 12 8 00 6 5 T

Consider the following set of vectors 1 4 V Let V V and v3 be column vectors in R3 and let A be the 3 x 3 matrix x V V V3 homogeneous linear system with augmented matrix A10 has a nontrivial solution Consider the following equation Solve for C C and C3 If a nontrivial solution exists state It or state the general solution in terms of the parameter t If only the trivial solution exists enter the trivial solution C C C3 0 0 0 C C C3 X Determine if the vectors V V and v3 are linearly independent The set of vectors is linearly dependent The set of vectors is linearly Independent with these vectors as its columns Then V V and v3 are linearly dependent if and only if the

Consider the following 00 0 1 00 0 0 32 A A A11 A12 A21 A22 A11811 A12B21 Compute each of the following using the indicated partitioning A11B12 A12B22 A21B11 A22B21 A21B12 A22B22 00 0 B11 B12 3 B21 B22 Compute AB by block multiplication using the indicated partitioning 32 1 1 0 0 00 0 0 1 1

Let and Ab Ab 2 A Use the matrix column representation of the product to write each column of AB as a linear combination of the columns of A B Ab 3 1 2 2 3 1 1 20 1 2 3 0 1 1 1 1 2 4 OOOH OOOH SCHOO Give the final product AB 11 000 000 000

Let 40 A 49 2 Compute the indicated matrix If this is not possible enter DNE in any single blank

Let 20 0 D 3 1 4 Compute the indicated matrix If this is not possible enter DNE in any single blank A A 3D 0 2 5 1

Consider the following set of vectors V 0 0 C1 V 5 V3 0 8 5 1 Let V V V3 and v4 be column vectors in R4 and let A be the 4 x 4 matrix V V V3 V4 only if the homogeneous linear system with augmented matrix A10 has a nontrivial solution Consider the following equation 0 0 0 Solve for C C C3 and C4 If a nontrivial solution exists state it or state the general solution in terms of the parameter t If only the trivial solution exists enter the trivial solution C C C3 C4 0 0 0 0 C C C3 C4 C4 0 with these vectors as its columns Then V V V3 and v4 are linearly dependent if and Determine if the vectors V V V3 and v4 are linearly independent O The set of vectors is linearly dependent O The set of vectors is linearly independent

Apply the Gauss Seidel method to the given system Take the zero vector as the initial approximation and work with four significant digit accuracy until two successive iterates agree within 0 001 in each variable Compare the number of iterations required by the Jacobi and Gauss Seidel methods to reach such an approximate solution Round your answers to three decimal places X1 X2 4 5X1 0 5x2 1 Xq 3 5X2 1

Let V V V be column vectors in R and let A be the nx m matrix the homogeneous linear system with augmented matrix A10 has a nontrivial solution Use the theorem above to determine if the set of vectors is linearly independent BOD O The set of vectors is linearly dependent O The set of vectors is linearly independent V V V V with these vectors as its columns Then V V Vm are linearly dependent if and onl

Consider the following set of vectors 4 0 0 0 V V3 V 1 Let V V and v3 be column vectors in R3 and let A be the 3 x 3 matrix V with these vectors as its columns Then V V and v3 are linearly dependent if and only if the homogeneous linear system with augmented matrix A10 has a nontrivial solution Consider the following equation 000 C C 3 C 1 4 4 Solve for C C and C3 If a nontrivial solution exists state it or state the general solution in terms of the parameter t If only the trivial solution exists enter the trivial solution C C C3 0 0 0 C C C3 Determine if the vectors V V and v3 are linearly independent O The set of vectors is linearly dependent O The set of vectors is linearly independent

Apply Jacobl s method to the given system Take the zero vector as the initial approximation and work with four significant digit accuracy until two successive iterates agree within 0 001 In each variable Compare your answer with the exact solution found using any direct method you like Round your answers to three decimal places X1 X2 7x1 X 54 X1 5X 2

8 H 4 8 for some x y Row reduce the associated augmented matrix Show that R2 span We must show that for any vector 1 1 ACCE R R R R So given a and b we have 10 we can write 01 0 2 DELC 1 1 R 2 DLL El 11 01

Using the boxplot find the following 2250 Part 0 2 Part 1 of 2 2400 2550 Maximum value 2700 2850 3000 Identify the maximum value minimum value median first quartile and third quartile

Rewrite 17 25 becomes 17 25 9 and as fractions with 20