Matrices & Determinants Questions and Answers

Practice Given each inequality and a set of possible solutions determine which values are solutions of each inequality Examples 1 and 2 1 12 t 7 19 18 17 2 h 4 2 1 1 2 3 7 a 10 4 3 2 5 8r48 4 5 6 7 4 4 3 2 ONLINE 4 18 15 d 2 3 4 6 15 23 40 45 50 8 18 2c 8 9 10

Cambridge has a job mowing lawns He mowed 5 lawns last week and earned 82 50 The table shows his earnings for the week Which coordinate pair represents the unit rate for this situation Cambridge s Earnings Number of Lawns Earnings 0 0 00 16 50 33 00 49 50 1 2 3 4 5 On 0 0 Ob 1 16 5 OL 3 49 5 66 00 82 50

Which expression is equivalent to O O 2x 8 13 5x 5r 8 2x 8 52 4 x 2 13 5x 2x 8 DONE 3 x 2 2 4 x 2 5

In Exercises 1 3 identify the initial amount a and the rate of growth as a percent of the exponential function Evaluate the function when t 5 Round your answer to the nearest tenth 1 f t 220 1 015 237 2 p t 5 5 1 5 31 8 3 h t 2 5 In Exercises 4 and 5 write a function that represents the situation 4 A college s tuition of 135 per credit hour increases by 5 each year 5 A bee population of 3000 increases by 40 every year In Exercises 6 8 identify the initial amount a and the rate of decay r as a percent of the exponential function Evaluate the function when t 3 Round your answer to the nearest tenth 6 f t 1420 0 895 4 x 2 6 8 10 15 20 y 5 7 y In Exercises 9 and 10 write a function that represents the situation 9 A 25 000 car decreases in value by 16 7 each year 10 A company s annual revenue of 487 000 decreases by 4 2 each year In Exercises 11 and 12 determine whether the table represents an exponential growth function an exponential decay function or neither Explain 12 11 X 9 2 y 8 y 9 2 5 1 9 13 81 54 36 24 In Exercises 13 15 rewrite the function to determine whether it represents exponential growth or exponential decay 13 y 0 3 14 y 3 1 6 15 y 9 0 68 3 16 A car has a value of 20 000 The value of the car is expected to decrease 10 annually for the next 8 years a Write a function that represents the value of the car in dollars y after t years b Use the function to estimate the value of the car after 4 years c Use the graph of the function to estimate the value of the car after 8 years

2 For each of the following functions find values a and b such that on interval a b the Inter mediate Value Theorem holds proving the function has at least one real root on that interval Graph the function on the interval to visualize your result Finally use Python to find the root on the given interval a f x 100e 100 0 012 b g x arctan a 1 e h x ln x x 100 2 Hint Note the domain of this function

6 What is an equation of the line that passes through the point 3 1 and is parallel to the line 4x 3y 15

Plot the image of quadrilateral ABCD under a reflection across line C l DO A B

Solve the system of equations 6x 6y 2z 34 3x 2y 6z 8 2x 4y z 13 ubmit your answer as an ordered triple x y z

Which element in the matrix A 5 2 1 2 O is the number zero

Multiply each pair of conjugates 2p 10w 2p 10w

A bathtub contains 120 gallons of water When the drain plug is pulled the tub drains 3 5 average rate of 8 gallons per minute In how many minutes will the amount of water in the bathtub be 24 gallons

b The system from example 2 2 3 Exercise 6 6 2 What goes wrong with the LU factorization if we use the row swap elementary operation Hint Thinl about what the corresponding E would be in contrast with the importance of the letter L a

n4 of 8 Step 1 any cubic decimeter

Find a general solution of the system x t Ax t for the given matrix A A x t 98 30 90 27 10 24 92 100 30

Find the reference number for each value of t a t 5m b d t t 51 4 37Tt 6 t 4

fit trend line and model C PRACTICE 1 Estimate the equation of the line of best fit for each graph y 20 18 16 8 6 4 2 0 1208642 YA 20 18 16 14 12 10 20 0 6 2 4 6 8 10 12 14 16 18 20 many points as possible but does not have to go through any of the points on the scatter plot You can use the equation for a line of best fit to make predictions about bivariate data through interpolation and extrapolation 2 4 6 8 10 12 14 16 18 20 A X X YA 20 18 16 14 12 10 8 6 4 2 0 YA 20 18 64 16 14 1086 12 10 to 8 4 2 OL 2 4 6 8 10 12 14 16 18 20 X 2 4 6 8 10 12 14 16 18 20 X

Consider the DTMC shown in the figure what s the probability measure of the union of the following cylinder sets Cyl S0S1 Cyl SoSsS6 Cyl SoSsS453 Cyl S0S156 A 2 3 B 1 3 C 8 9 D 2 9 72 5 S 14 S Show Transcribed Text Question Question What is the probability of reaching the set of state sa

1 a Determine if b is a linear combination of V V2 V3 A 1 V3 6 b V 1 5 0 2 8 b Construct a 5 x 5 matrix A and vectors b and c in R5 so that Ax b has a solution Ax c does not 6

2 B 3 C 1 YD Si A 2 Calcula a 2A 3B b A C 3B 4A c D 3A 5B C 0 5 6 2 7

For each given matrix A first compute det A Then interchange two rows of your choosing and compute the determinant of the resulting matrix A 0 1 1 1 31 2 04 a A det A det A b A det A det A 24 1 01 1 00 4 Form a conjecture about the effect of row interchanges on determinants O Interchanging rows doubles the determinant Interchanging rows does not change the determinant O Interchanging rows halves the determinant Interchanging rows changes the sign of the determinant Interchanging rows has no consistent effect on the determinant

Choose the true statements about vectors Select one or more a Velocity is an example of a vector b Vectors have both direction and magnitude c Vectors can be single or multiple dimensional d x y is one way to express the components of a vecto e Vectors are expressed as a line

Set up the definite integral required to find the area of the region between the graph of y 18 x and y 4x 42 over the interval 4 x 5 S dx

For the function h x 5x 12x a h 4 b h 3 h 2 1 find

Use the given information to prove that ADEF ADHG H 1 D Statement Validate Given ZDEF ZDHG Prove ADEF ADHG Reason Reason Send To Proof Send To Proof Dia 11 H

Question 1 Write the following systems of linear equations using matrix notation 1 2 I1 I2 I3 5 2x12x2 6x3 1 4x1 5x2 6x3 8 x z y 3 5

If circle passes through the centre O of circle 6 the length of the common chord AB is equal to the tangent segment AC R H EDDY MEMORIAL UNIVERSITY OF NEWFOUNDLAND ST JOHN S NEWFOUNDLAND CANADA AIC 5S7 ovide a formal proof of the statement Note that you cannot assume the labeled congruent parts are

5 You are given 1 2 01 3 205 A 4 6 15 5 3 A 7 5 2 3 2 6 2 1 3 1 2 0 1 0 5 11 a Calculate the inverse of D b Find the solution of the equation Ax B c Compute det C6 B 3 C 11

Use the following matrix A 3 4 d 2 C 5 5b a 2 and the following values a 0 2 b 2 4 c 4 1 d 1 5 Compute the inverse of A and indicate its entry at row 2 column 2

In Exercises 67 74 find the matrix A given the reduced row echelon form R of A and information about certain columns of A 204 a2 0 0 71 R and a3 1

Use matrix inversion to solve the collection of systems of linear equations a X X X X x y z b X X X X 2y X x y z c 4y 2z 10 11 14 X V z y 4y 2y y y 4y 2y N N 2z N N N N 6 1 2z O 0 0 0

Match the addition and subtraction problems to their solutions 1 5 3 77 24 1 3 6 1 15 3 6 1 4 1 2 A B C D 27 2 3 10

Find the vector equation for the line passing through the point P 6 5 10 and parallel to the line x 9 10t y 3 t z 4 4t x 814 y 0 0 0 0 0

periods of y 4 tan 3x Make sure to include asympotetes of x y intercepts Answer s 2n T 2 kic 203 x 3r x 2x

Determine if the following statements true or false with proper justifications a If A is a square matrix then A AT A AT b If A and B are n x n invertible matrices and A is also invertible Then A is invertible c If A B and C are square matrices such that AB AC If A is investible then B C d If a square matrix A satisfies A5 A A I then A is invertible

Exercise 4 14A Algo Single step and multistep Income statements LO 4 6 The following Information was taken from the accounts of Green Market a small grocery store at December 31 Year 2 The accounts are listed in alphabetical order and all have normal balances Dollar amounts are given in thousands Accounts payable Accounts receivable Advertising expense Cash Common stock Cost of goods sold Interest expense Merchandise inventory Prepaid rent Retained earnings 1 1 Year 21 Sales revenue Salaries expense Rent expense Gain on sale of land Required Prepare an Income statement for the year using the single step and multistep approach Complete this question by entering your answers in the tabs below Single Step 601 401 200 411 201 600 69 440 41 636 1 010 148 100 46 Multistep Expenses Prepare an income statement for the year using the single step approach Enter values in thousands of dollars Amounts to be deducted and losses should be indicated with a minus sign GREEN MARKET Income Statement For the Year Ended December 31 Year 2

Find an LU factorization of the given matrix 33 1 954 34 21 4

9 IfT P M 2 is a linear transformation and 7 2 x 1 7 3 x T T 1 then find 7 x 2x 2 31 a b 1 2 27 2 D 1 Ans a an

Recalling that the spectral radius of a square matrix A is p A max A build the Hilbert matrix A of order n 14 Starting from the vector with all elements equal to 1 apply 4 iterations of the power method to compute an approximation of the spectral radius p A Then use the Matlab function eig to compute the reference exact value The relative error associated to the approximation of the spectral radius computed by the power method is approximately 9 2136e 07 X 4 8734e 08 2 4567e 10

Show that the set is a subspace of R S x R 7x1 6x2 x3 6x 3x2 x3 0

Find the missing parts 6 marks e t p W Side w 5 36 m and p 6 78 m sin 0 0 sin tan o t For each answer below round to 3 significant digits Include units where appropriate

Use an inverse matrix to solve each system of linear equations a X X X1 8x3 0 X28X3 X1 8x2 x3 0 X1 X2 X3 5 X1 X2 X3 1 b X1 X2 8x3 7 X1 8x2 x3 2 X1 0 X2 X3 X1 X2 X3 00

1 Find the inverse of the relation 1 0 2 1 3 0 4 2

work graph of y f x gives the speed limit y along a country road r traveling x miles What are the highest and lowest speed limits along this stretch of Estimate the miles of road with a speed limit of 65 mph Evaluate f 4 and f 8 Maximum MusouvIT TV GVJI Part 1 of 4 What are the maximum and minimum speed limits along this retch of highway Im mph HI

basis for the column space of A is vector A 2 4 2 2 6 4 6

Use the graph of the quadratic function f to determine the solution a Solve f x 0 b Solve f x 0 a The solution to f x 0 is Type your answer in interval notation b The solution to f x 0 is Type your answer in interval notation 15 10

2x y 4z 6 Ive the system 6x 5y 11z 9 by using an LU factorizatio 2x 3y 92 7

Consider the linear transformation T R R where a 6 0 0 0 T 2 7 T T 1 3 Find a matrix A such that T v Av Enter the matrix A in the response area below To enter a matrix click on the button with 9 dots arranged in a square and choose the number of rows and columns Then edit the entries of the matrix If you accidentally mess up the format of the matrix click the butt with the rubbish bin to reset the entry area 11

EX 1B Calculate the table G n 2 where G 1 0 0 X y Z 1 U V 0 1 W 0 0 0 0 and to continue to be applied for n 2023 kai x 1 y 2 z 0 u 2 v w 3 22 A B An KBK write the matrix G in the form G I Q and then apply the binomial expansion for commutative matrices that i AB BA then

2 points Let A be a mxn matrix Show that for all b in Rm we have that AA b projcol 4 and for all a in R we have that A Ax projcol AT T

then A 1 Given b x 3 25 solve Ax b 3 A 5