Matrices & Determinants Questions and Answers

Find a basis and the dimension of the solution space of the homogeneous system 2x2 x3 2x4 x5 0 1 2x2 2x3 x4 0 x1 4x2 x3 2x4 x5 0

Y x y z b x 4y 22 7 X 2 5 x y 2 4 x y z c x 4y 2z 0 X 2 0 x y 20 z 1 z 7 x y z Submit Answer 0 1 Points 0 0 0 A Viewing Saved Work Revert to Last Response DETAILS For this exercise let PREVIOUS ANSWERS and C WANEFMAC7 5 R 003 01 Determine whether the expression is defined If it is evaluate it If an answer is undefined

Solve the system If the system has an infinite number of solutions parameterize them by letting z t 3x x X X y Z t 4y 4y 2y 15z 25 13z 19 7z 11

1 a b Let A 0 2 c 00 1 iagonalizable Find the condition on a b c for which A is AX XA A X XA

A 3 1 03 00 1 0 0 a Show that A 5A 3A 91 0 b Use part a to show A has an inverse aka is non singular and compute A

Consider the subspace V of R given by 3 0 4 1 a Find an orthonormal basis for the orthogonal complement V V span

2013 5 2014 6 2015 10 a Determine the polynomial function of least degree whose graph passes through the given p X p x

2 The area of a circle is 64 7 inches squared What is the circumference of the circle in terms of 7

6 Consider the underdetermined linear system 3y 2 w 3 2r 2y 2 2 8 3x y 23 3011 Find three vectors 1 m and n in R such that the solutions to the system have the form Al where A R are free variables

JE DE 32 30 Determine if the set S p2x2 is span set of

For the matrix A find if possible a nonsingular matrix P such that P AP is diagonal If not possible enter IMPOSSIBLE 2 0 0 5 5 1 1 0 5 P Verify that P 1AP is a diagonal matrix with the eigenvalues on the main diagonal P AP

1 point Solve the system associated with the augmented matrix below If the system is inconsistent type No Solution in each blank If the system is dependent use the variable z as your free vari 10 2 13 7 13 6 5 4 11 4 6 12 The solution to the system is Ex y z

3 Let A A2 A3 be the columns of a 3 x 3 matrix A such that A A2 7A30 Is it true that A is singular

Use row reduction to determine which of the following matrices have an inverse For those that do have an inverse find it a b NO C 5 3 1 4 3 2 11 1 1 1 2 2 0 d 2 3 1 4 3 2 1 13 1 31 Note The matrix in 2 d appeared in the last assignment 1 1 0 0 0 1 1 1 1 2 2 3

7 Multiply each expression by k 9 k In which expressions does the entire denominator cancel a b C d e 7 9k 2k 9 k 414 k 2 5 k k 13 k 9 k

f x 4 for 1 6 for x 1 Find f 0 al

A bag contains 3 red 5 yellow 2 green and 6 blue marbles If one marble is chosen at random replaced and then a second marble is chosen at random find the probability of obtaining 2 marbles of different colors

A vegetable farmer sells three different vegetable packages The three packages are made up as follows Package A 1 cucumber and 3 tomatoes Package B 1 broccoli 2 cucumbers and 2 tomatoes Package C 2 broccoli and 2 cucumbers Suppose you would like to order some number of each of the three packages so that you ultimately get 20 broccoli 40 cucumber and as few tomatoes as possible a Let a represent the number of tomatoes you end up getting from your order Set up a system of equations whose solution would determine the number of each of the three packages you should order to get 20 broccoli 40 cucumber and a tomatoes b Now write down the augmented matrix associated with the system of equations from a c Use elementary row operations to obtain the RREF of the augmented matrix from b d What is the smallest number of tomatoes you will have to buy in order to get 20 broccoli and 40 cucumber Explain your answer

et a 1 3 1 and z 1 2 3 Write down the dot product 7 7 as a matrix multiplication The solutions of Az 0 are the vectors in R that are perpendicular to the vector raw the row and column pictures for the equations z 3y 4 and x 2y 1 se backward substitution to solve the following two systems of equations 2 1 22 and 21 2 3x3 202 323 6

For a b R the transformation T R R defined by T x y 2x y 3x y 3a 5x 2y bxy is linear if 1 a b 1 2 a 0 b 1 3 a 1 b 0 4 None of the giuon mora in true

a a b d e C a b 5 2 6 1 3 8 1 6 2 5 8 3 348 248 8 8 2 Determine whether each equation is true or false and place a mark in the appropriate box If the equation is false change one value of the equation to make it true and write the altered equation on the line True False multiplication expression with its equivalent addition expression 2 2 2 2 2 3 6 18 5 2 10 3 2 Date 1 0 03 0 03 Period 5 5 1 1 1 1 1 1 x26 3 3 3 3 3 3 3 3 Altered Equation 3x6 18 Keep Chanse Fill 5 2 5x2 16 5 223X 3S 3 811003 0 03 3 Shawn and Jada are walking in the same direction on a path through the park They walk by the fountain on the side of the path at the same time When they pass the fountain Jada is walking at a speed of 5 ft second and Shawn is walking at a speed of 3 5 ft second If 0 represents the location of the fountain what values represent the location of each person in 5 seconds Show or explain your thinking b 8 seconds before arriving at the fountain how many feet in front of Jada was Shawn Show or explain your thinking

7 6 to ex 27 H 45 4661 What is the least possible degree of the polynomial graphed above

If A E C preserves length A 2 2 for all x E C then A is unitary Give an alternative proof using the SVD

2 1 3 2 3 3 3 2 24 Find the eigenvalues and eigenvectors of A 1

21 Show that 3 3 0 2 2 1 1 1 4 is an orthogonal basis for R with respect to the standard inner product Find the coordinate matrix of 5 3 1 relative to the above basis

39 Let T be the linear operator of the vector space V R defined by T 21 22 23 2r2 33 2x1 x2 2x3 5x2 423 Find a basis for ker T and a basis for Ran T

What is the domain of the function in the graph 20 18 16 14 12 10 8 649 2 72 to 00 g 5 00 f 5 011 f 16 011 g 16 678 ob

Use Figure 9 2 to complete problems 2 and 3 Check your answers in the back of the book 2 Find the average wholesale price for a used V6 four door sedan with air conditioning leather seats front bucket seats power sunroof and aluminum wheels It has 75 500 miles

T Z Amplitude Period Phase shift 1714 1 4 0 5 0 5 1 5 0 2 5 6 What is the equation Amplitude 0 Period Phase shift 2 3 m4 m 4 1 2 11 2 3TT 4 31 4

3 Assume ex ey ez is an orthonormal vector basis The following vectors are defined a 4x 3y ez b 6ey ez c 8ex ez Are a b and linearly independent If not what is the relationship between the vectors

A straight line is defined by the following linear equation 3y 2 x 1 5x 7 a What is the slope of the line b What is the y intercept of the line

Write an augmented matrix ror tre syorc 4x 7y 9z 4 0 3x 5y 4z 8 0 3y 2z 1 0

Question 5 Assume that W and M are subsets of a vector space V Show if the following statement is ALWAYS correct or not If W is included in M and M is independent then W is independent

Which of the following set of vectors span R A 1 2 1 2 1 3 2 4 2 B 2 2 2 1 0 1 3 2 3 C 1 0 0 0 1 0 0 0 1 1 1 1 D 1 1 1 1 0 1 E None of the above

1 Which of the following set of vectors span R A 1 2 1 2 1 3 2 4 2 B 2 2 2 1 0 1 3 2 3 C 1 0 0 0 1 0 0 0 1 1 1 1 D 1 1 1 1 0 1 E None of the above

Q3 Find the kernel of the linear transformation T x y x 2y y x T x y x y y x a T R R b T R R

Consider the equation 3x 5y 15 A line parallel to the above line would have a slope of A line perpendicular to the above line would have a slope of

Use the Gram Schmidt process to produce an orthonormal basis for the given set 4 4 4 2

b The definition of observability can also be applied to nonlinear systems Is the nonlinear system X1 12 X1 2 6 x2 0 3 X2 x2 y e observable

a Describe the range in w plane of the following function f z 4 mark

Simplify 4 3 2 5 6 2 O 32 30 6 15 10 6 8 8 24 5 8 15 24 6 E

A normally distributed data set has a mean of O and a standard deviation of 0 5 Which is closest to the percent of values between 0 5 and 0 5 O 68 O 95 O 34 O FOR

stion 1 Determine whether the set GIED is linearly independent 20 Point

5 Ali plans to go to college in a year and needs to save for tuition He invests his summer earnings of 3050 part at 8 interest per year and part at 7 5 per year After one year he has earned a total of 234 in interest How much did he invest at each rate 6 Abby mixes cinnamon and nutmeg to make 25 g of a spice mix Cinnamon costs 9 cents g and nutmeg costs 12 5 cents g The spice mix costs 9 7 cents g How much of each spice does Abby need to use ANS C 20 and N 5

sing the laplace Expansion method find the determinant of the matr 459 13 2 6 4 14 3 7 4 15 48 4 161

Let L R R be a linear transformation for which we know 1 2 LI O 0 6 3 1 Then matrix of linear transformation is 2 3 1 A 3 4 2 1 1 0 Select one True 3 1 L 0 0 8 6 4 L 2 1 0 False

A car depreciates at a rate of 6 Lisa purchased a car for 23 000 Using the appropriate decay factor how much should it be worth 10 years after Lisa purchased it 3 833 33 12 388 15 23 060 00 41 189 50 Question 5 14 3 points A culture of bacteria doubles by the end of each hour If there were 50 bacteria in the initial culture how many will there be after 1 hour has passed 100 bacteria 60 bacteria 52 bacteria 25 bacteria Question 6 14 3 points Determine the value of th

using row elementary operation transform the basis 1 0 1 1 0 1 0 3 5 of R3 to obtain an orthonormal basis

a 2 b 4 c 4 Let B a b 1 c 3 b c 2 a 2 c a 3 b 1 0 1 0 be a 3 3 matrix Obtain QR decomposition of the resulting matrix B possible always

Suppose that p 9 and are three non coplanar vectors in R Let the components of a vector 5 alc p and 7 be 4 3 and 5 respectively If the components of this vector 5 along p a r p q and p q r are x y and z respectively then the value of 2x y z is