Matrices & Determinants Questions and Answers

Problem 5 Let W be the subspace of R4 spanned by the vectors u 1 0 1 0 u 0 1 1 0 and u3 0 0 1 1 Use the Gram Schmidt process to transform the basis u U U3 into an orthonormal basis 6 4 v 2 0 2 0 v V6 V6 V6 0 v3 V2 3 B V 2 0 2 0 v2 6 V6 6 0 v3 3 v 3 3 66 13 131313 6 6 2 1 2 0 2 0 v 6 V6 V6 0 v3 V2 3 Problem 5 Select 2 0 12 0 v 6 V6 V6 0 v3 3 V2 3 E v 2 0 2 0 v2 V6 V6 V6 0 v3 D v 2 0 V2 0 v2 V6 V6 V6 0 v3 3 V3 V V V2 3 6 3 3 3 3 6 6 6 F v 2 0 2 0 v2 V6 6 6 0 v3 3 3 3 3 3 6 6 3 3 3 6 6 6 G H v 2 0 2 0 v 6 6 6 0 v3 V2 V6 6 3 3 3 3 3 6 3 V3 3 6

Problem 10 a Problem 10 b A 1 2 5 4 Just Save 1 2 3 0 a Find the dimension of the row space of A b Find the dimension of the nullspace of A Problem 10 Your Answer 3 3 5 6 7 9 1 0 Submit Problem 10 for Grading Attempt 1 10 a 10 h Attempt 2 10 a 10 h Attempt 3 10 a 10 h

Jeremy can run no more than 1 5 miles every 10 minutes At this pace Graph the Inequality to represent how many miles Jeremy can run in 1 hour 1 5x10 15 15x6 90

p exceeds 24 24 38 0 38 25

Problem 8 Let W be the subspace of M12 12 i e 12 x 12 matrices consisting of all matrices whose diagonal entries are zero Find the dimension of W Problem 8

Find an orthogonal matrix A where the first row is a multiple of 0 1 0 A

Problem 7 Find a basis for the subspace of R4 consisting of all vectors of the form a b c d where c a 7b an d a 5b Problem 7 A 0 1 7 1 1 1 0 5 D 1 0 1 1 0 1 7 5 G 1 0 7 1 1 0 1 5 Select B 1 1 0 1 0 1 7 5 C 1 0 7 1 0 1 1 5 F 0 1 7 1 0 1 1 5 E 1 0 1 1 1 0 7 5 H 1 1 0 1 1 0 7 5

roblem 1 Which of the following sets of vectors are independent i 1 2 1 3 2 0 2 4 1 ii 1 2 1 3 2 0 5 6 0 iii 1 2 1 3 2 0 7 10 2 Problem 1 A none of them B ii only C i only D i and iii only E ii and iii only F iii or G all of them H i and ii only F

Solve the given system of equations If the system has no solution say that it is inconsistent x 2y 3z 13 2x y z 1 3x 2y 2z 13 Select the correct choice below and fill in any answer boxes within your choice OA The solution is x y and z Type integers or simplified fractions OB There are infinitely many solutions Using ordered triplets they can be expressed as x y z x y z any real number Simplify your answers Type expressions using z as the variable as needed OC There are infinitely many solutions Using ordered triplets they can be expressed as ou

11 ABC geni gen DBE ye benzer B l m A X in de erini bulmak i in bir denklem kurun ve z n A 4x 3 7x 30 B E

Problem 3 Which of the following sets of polynomials form a basis for P2 P1 1 3x 5x P2 5 x ii P 1 x P2 10x x P3 1 11x x iii P 1 P2 1 6x P3 x P4 x Problem 3 A all of them B i only C i and ii only D iii only E none of them F ii and iii G i and iii only H ii only Select

Problem 2 Let p 8x 6x 3 Find the coordinate vector of p relative to the following basis for P2 P1 1 P2 1 x P3 1 x x Problem 2 Enter the coordinates separated with commas

a Find the center radius form of the equation of the circle with center 3 5 and radius 4 b Graph the circle a The center radius form of the equation of the circle is x 3 y 5 16 Type an equation b Use the graphing tool to graph the circle Click to enlarge graph 4 10 B Ay 10 8 6 4 2 2 8 40 x

Convert the Cartes in coordinate 6 4 to polar coordinates 0 0 2 r Preview

Rationalize the denominator 3 8 2 3 3 5 2 3 8 2 3 3 5 2

4 Find the eigenvalues and eigenvectors for the matrix 35 3 046 001

Solve the following logarithmic equation Express irrational solutions in the exact form and as decimals 2 log 3 x 1 3 log 35 log 325 www Use a comma to separate answers as needed Simplify your answers Type exact answers using radicals as needed Express irrational solutions as decimals Select the correct choice below and if necessary fill in the answer box to complete your choice OA The solution set is Use a comma to separate answers as needed Round to three decimal places as needed B The solution set has no irrational solutions

Solve the system State whether the system is inconsistent or has infinitely many solutions If the system has infinitely many solutions write the solution set with y arbitrary 5x 6y 1 10x 12y 2 Select the correct choice below and if necessary fill in the answer box to complete your choice OA There is one solution The solution set is Simplify your answer Type an ordered pair OB The system has infinitely many solutions The solution set is Simplify your answer Type an ordered pair Type an expression using y as the variable OC The solution is inconsistent

Use the vectors in the figure at the right to graph the following vector 2v u 3w Select the figure with a vector that is equivalent to 2v u 3w O A B D

Find the area of the triangle ABC C 133 8 a 46 3 ft and b 36 4 ft Points 0 of 1 The area is square ft Simplify your answer Type an integer or decimal rounded to the nearest tenth as needed

1 Do the following a Solve each equation for x Give ex log5 125 x log 4 1 log25 x

Click here to watch the video Given that cos 1 A 5 5 and 0 x 2 determine the exact value of cos a cos 2 Simplify your answer including any radicals Use integers or fractions for any numbers in the expression Ratic

Test Find the 18th term of the arithmetic sequence 4 9 14 19 The 18th term is Ques

A ZA LB LC b degrees a degrees degrees 5 b 6 c 9

You approach a hill on top of which there is a tall radio antenna You know from your map that your horizontal distance from the bottom of the radio antenna is 600 feet The angle of elevation to the bottom of the antenna is 10 and the angle of elevation to the top of the antenna is 25 You figur that the height of the hill is feet and the height of the antenna is feet Enter your answers rounded to the nearest foot Hint Draw a picture Figure out the height of the hill Figure out the combined height of the antenna and the hill Compute the difference

Find the area if the distance between two adjacent dots in a row or column is one unit 9 What is the ratio of the area of B to the area of A 3 A B 9 19 D 1

A researcher wants to know if the news station a person watches is a factor in the amount of time in minutes that they watch The table below shows data that was collected from a survey Local FOX CNN 55 42 Ho H1 H2 H3 H At least two of the means differ from each other 1 For this study we should use ANOVA 2 The test statistic for this data 3 The p value for this sample 37 65 25 53 65 24 29 51 47 35 26 63 62 40 50 Assume that all distributions are normal the three population standard deviations are all the same and the data was collected independently and randomly Use a level of significance of a 0 01 49 69 17 46 60 17 46 Please show your answer to 3 decimal places Please show your answer to 4 decimal places

Use properties of logarithms to find the exact value of each expression Do not use a calculator log 86 8 log 26

2 Is x 4 a solution to the equation 4x 5 21 No

Rewrite the rational exponent expression as a radical expression O The expression cannot be written as a radical expression

18 T B P 10 2 0 10 2 0 b 10 Find the coordinates of x in the ordered basis B a b Hint The preview shows the picture of a with the coordinates you pr

Let A and B be any two sets Then a AUB ANB A b ANB AUB De Morgan s Properties A B B U U 11 U A B A U B D

Determine which quadratic equations have 2 real solutions Select two that apply 6x 6x 3 0 9x 3x 9 0 4x 8x 4 0 2x 10x 5 0

1 2 Evaluate each determinant 7 12 6 26 29 32 35 T 5 1 6 603 3 1 6 2 5 4 2 3 5 7 6 8 ble 5 2 6 5 0 4 9 6 8 10 27 41 4c 5d 39 5c 8 54 d 55 30 8 11 9 12 2 0 3 4 2 5 8 3 0 0 0 33 8 2 8 36 2 NG c c c 5 5 3 4 2 a 28 Use Cramer s Rule to solve each system of equations 39 6x 5y 73 7x 3y 71 31 5 LEA AND 0 444 1 14 ON Ada bat ba 5 1 2 Extra Practice is on page 34 4 6 9 2 5 8 WO DON DAN averaged 38 TRAVEL Mr Smith s art class took a bus trip to an art museum The bus 65 miles per hour on the highway and 25 miles per hour in the city The art museum is 375 miles away from the school and it took the class 7 hours to get there Use Cramer s Rule to find how many hours the bus was on the highway and how many hours it was driving in the city bits ab oib 1 2 10 12 40 10a3b 34 3a 8b 28 37 6 0 13 42 6f 8g 22 11f 5g 60 44 11 772 4

13 K 1 3 17 20 4 15 P 38 Q Find the inverse of each matrix if it exists 7 8 3 0 21 whether each pair of matri 3 2 1 2 L 0 1 1 3 6 1 5 6 31 4 9 2 3 1 5 18 21 24 2 3 3 2 5 4 17 4 3 2 7 4 2 2 8 14 M 16 R

6 8 1 33 2 4 2 4 1 2 316 0

Find a matrix P such that PTAP orthogonally diagonalizes A Verify that PTAP gives the proper diagonal form Enter each matrix in the form row 1 rov comma separated list A 0 10 10 0 10

47 DOUGHNUTS Mi Ling is ordering doughnuts for a class party The box contains 2 dozen doughnuts some of which are plain and some of which are jelly filled The plain doughnuts each cost 0 50 and the jelly filled cost 0 60 If the total cost is 12 60 use Cramer s Rule to find how many jelly filled doughnuts Mi Ling ordered

Orthogonally diagonalize the matrix giving an orthogonal matrix P and a diagonal matrix D To save time the eigenvalues are 10 2 and 2 442 A 424 244 Enter the matrices P and D below Use a comma to separate answers as needed Type exact answers using radicals as needed Do not label the matrices

1 point Find a linearly independent set of vectors that spans the same subspace of R as that spanned by the vectors A linearly independent spanning set for the subspace is 1 2 0 0

1 3 Divide 1 3 2v 2 v 6 A 2 v 3 2 v 4 2 v B v C v 3 3 D 1 2 v 4 3 3n n n 5 2 n A n n 3 1 B n 2 C n 2 D n 1 2 11 2 2 n 3 2 n 5 2 n 4 2 2 2n 3n 2 n 1 A 2n n 1 B 2n n 2 C 2n n 3 D 2n n 3 3 n 1 3 n 1 3 1 1 4 8 2v 11 10v 2 v A 8 5 2v 1 B 8v 5v 2 C 8v 5 2v 5 1 2 v D 8v 5v 4 2 v 4 2 1 2 v

3 3 3n n n 5 2 n A n n 3 B n 2 C n 2 1 D n 1 2 n 2 2 n 2 n 5 2 n 2022 Cara Sefware LLC All rig 4 4 8 2v 11 10v 2 v A 8 5 2v 1 7 2 v 4 2 v B 8 5 2 C 8v 5v 2v 2v 5 D 8v 5 4 2 2 v 2 v Made with infinite Alethra 2

2 v Using Cayley Hamilton theorem find A for A 2 et A be any non zero vector square in ma Said

9 Use the sets U following set a d e f a b c d e f A a b c B a c f to list the elements of the d An B ANB de b B B b d e e An B 10 Use a Venn diagram to illustrate the set AUB nC A We can use Venn Diagrams to identify sets of data and show the equalities of various sets Eventually we will solve problems and identify probabilities using Venn Diagrams Let s do some examples II Use a Venn diagram to illustrate the following equality B C AUB AUB de U f AUB AUB ANB U ANB U ANB A A B B U

Let v 0 2 3 and v 3 1 0 V1 Select all vectors v3 such that V V2 V3 is a basis of R A V3 2 3 2 B V3 3 1 3 3 1 0 C V3 D V3 0 0 0 OE V3 3 3 3

10 3 2 0 Find the exact value of tan Given sin0 A 41 41 B 41 C 41 41 D 41 41 21 and 0 they lost a bet it was their friendship anniversary they had nothing better to do they wanted to do something fun

Find a singular value decomposition A PDQT where A 0 88 Enter the matrices P D and QT below Any empty entries are assumed to be 0 P 1 QT 1 1

oblem 10 Which of the following sets are closed under addition i The set of all vectors in R of the form a b where b a ii The set of all 3 3 matrices that have the vector 1 3 2 as an eigenvector iii The set of all polynomials in P of the form ao a x a2x where ao A iii only B ii only C i and iii only D i only E i and ii only F ii and iii onl G all of them H none of them

oblem 11 Let v 1 2 1 and v 3 2 0 Which of the following vectors are in span V V i 2 4 1 ii 5 6 0 iii 7 10 2 Problem 11 A all of them B none of them C i and iii only D i only E ii and iii only F ii only G iii only H i and ii only Select v

Exercise 13 5 5 Verify the following diagonalizations starting from the characteristic polynomial solving for the eigenspaces and then inverting the matrix of eigenvectors A 1 B 22 1 0 1 0 2 1 2 X X 0 1 0 0 1 3 1 1 1 2 1 C 2 4 2 3 6 3 1 YA Y ZACZ 1 1 3 2 1 2 5 3 673 6 96 0 0 00 1 1 0 1 1 1 3 2 3 1 3 2 1 2 1 3 1 1 2 0 1 1 0 2 00 0 10 0 02 200 000 000 12 5 3 9 8 0 3 3 6 3 6 2 2 3 5 2 0 3