Matrices & Determinants Questions and Answers

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

lem 5 Find the parametric equations of the line in R that passes through the points 5 5 2 and 1 4 9 B x 6 6t y 5 2t 2 11t C x 5 6t y 5 t 2 11t E x 5 6t y 5 A 6 6t y 5 t 2 12t F x 5 6t y 4 t 2 11t G x 6 6t y 5 t 2 12t H x 5 5t y 5 t 2 11t D x 5 6t y 5 t 3 11t 2 12t

inner products T a u v where u 12 T v 5 b x y where x 10 14 3 y 0 49 00 c 2u v v using u and u above Compute the norms of u v x and y above simplify only where reasonable Compute the distance from u to v above Find the angle between u and u above calculator needed Draw two vectors in R2 which are not perpendicular Verify this by showing that a their dot product nonzero and b the Pythagorean for dot products does not hold for these vectors Prove that the standard dot product we re using actually satisfies all of the properties of an inner produc

e 13 5 1 Diagonalize 1 0

1 Let A 0 0 Select all of the statements that are true B 0 and C 0 4 OA The matrix Cis a diagonal matrix B The matrix product AB is defined OC The matrix sum A C is defined OD The matrix Cis a square matrix OE The determinant det A is defined OF The scalar multiple B is defined OG The matrix B is an identity matrix OH The matrix power C is defined OI The matrix product AC is defined 0 83 3 0

a Find a matrix B so that BA A for any 2 x 4 matrix A B b Find a matrix C so that AC A for any 2 x 4 matrix A C

The truss is made of three A 36 steel members each having a cross sectional area of 206 mm Determine the magnitude P in kN required to displace the roller to the right 0 368 mm 0 8 m 0 8 m B P 5 kN 0 6 ms

Let A 27 3 6 Answer R Find an example of a nonzero 2 x 2 matrix B so that AB a 2 9 0 c d Hint Try finding a b c d so that A SED

DA If C M are n x n matrices then C M C 2CM M B If C is a square matrix so that C2 0 then C 0 OC If C Mare n x n matrices so that CM 0 then C 0 or M 0 OD For any n x n matrix C there exists a matrix M so that CM MC DE If C Mare nx n matrices so that CM 0 and M has an inverse then C 0 OF For any n x n matrices C M it s true that CM MC

7 Find all values of k such that the following system of equations x 2y 3z 9 2xy z 8 x 3y k 1 z k has i Unique solution ii More than one solution and find them iii No solu

A faucet is used to add water to a large bottle that already contained some water After it has been filling for 5 seconds the gauge on the bottle indicates that it contains 18 ounces of water After it has been filling for 13 seconds the gauge indicates the bottle contains 58 ounces of water Find and interpret the average rate of change in the volume of water per second OA The average rate of change is 6 This means that the volume of water is decreasing at a rate of 6 ounces per second OB The average rate of change is 6 This means that the volume of water is increasing at a rate of 6 ounces per second OC The average rate of change is 5 This means that the volume of water is increasing at a rate of 5 ounces per second OD The average rate of change is 5 This means that the volume of water is decreasing at a rate of 5 ounces per second

Solve the equation 4 0 5x x X 5 The solution set is 6 Simplify your answer

Determine if each of the subset W is a subspace of the given vector space V a 1 0 a b eR V M22 with standards operations 0 b b W a b a b 2a a b eR V R with standards operations a W

out of the microwave and has an initial temperature of 198 F The drink is placed on the counter in a room whose temperature is 73 F After 7 minutes the temperature of the hot chocolate has dropped to 150 F a Find the function T t which gives the temperature of the hot chocolate in degrees Fahrenheit t minutes after the hot chocolate was removed from the microwave T t 73 125e 0 069t Not quite 113 82 F b Estimate the temperature of the hot chocolate 16 minutes after the drink was removed from the microwave Round to two decimal places That s not it Preview T t 73 125e 0 069t oF F Preview 113 82 F c Estimate the number of minutes after removal from the microwave for the temperature of the hot chocolate to reach 91 F Round to two decimal plago

Apply Cramer s rule to solve the following equations x 3y 6z 2 3x y z 9 X 4y 2z 7 Select one O a X 1 y 2 z 1 O b X 2 y 2 z 1 Oc X 2 y 1 z 0 5 O d X 1 y 2 z 0 5

Find the LU factorization of 2 2 2 12 15 14 4 13 16 That is write A LU where L is a lower triangular matrix with ones on the diagonal and U is an upper triangular matrix A 5 25 A 10

Determine by inspection whether the given set of vectors is linearly dependent of independent Do NOT perform row reductions of a matrix Justify your answer 4 22 3