Matrices & Determinants Questions and Answers

5 marks Consider the vector space R with basis Find the dual basis 2 0 0 2 2 3 2

1 point Find the matrix A of the reflection in the line L in R that consists of all scalar multiples of the vector 19 A 1 12 37

vector 12 is a linear combination of the vectors 2 and 6 if and only if the matrts equation AS has a nomion where 14 1 1 1 1 and 1

6 Consider the vector space M2x2 equipped with the standard inner product A B tr B A Let 1 2 I If Wspan A B then what is the dimension of the orthogonal complement W A 0 B 1 C 2 D 3 E 4 A 99 1 2 and B

1 point Let Find bases for the kernel and image of T z Ar A basis for the kernel of Ais A basis for the image of Ais 1 2 1 40 19

Let v m 5 5 V2 1 Is w in v1 v2 03 Enter yes or no 20 3 20 V3 51 8 50 and w 2 How many vectors are in v1 02 03 Enter inf if the answer is infinitely many 3 How many vectors are in Span v1 02 03 Enter inf if the answer is infinitely many 4 Is w in the subspace of R spanned by 01 02 03 Enter yes or no I 6 6

Suppose I is the transformation from R to R that results from a y expansion by 5 followed by a rotation by T Find the matrix A that induces T 0 0 0 0 A

fog 4 fog 3 LA gofi 0 3 Drag and drop options on t right hand side and submit Forkeyboard navigation SHOW MORE g 1 4 2 14 2 E 4 3 1 5 y g x y f x

Question 4 Sketch a graph of y 4 8 7 8 7 6 5 4 3 2 6 S 32 3 2 1 2 3 4 5 6 Clear All Draw 7 8 1 2 3 4 5 6 7 8 A AV

The population of a certain country from 1970 through 2010 is shown in the table to the right a Use your graphing utility s exponential regression option to obtain a model of the form y ab that fits the data How well does the correlation coefficient r indicate that the model fits the data b Rewrite the model in terms of base e By what percentage is the population increasing each year The model of the form y ab that fits the data is y Type integers or decimals rounded to three decimal places as needed How well does the model fit the data Since the correlation coefficient is r Round to three decimal places as needed the model fits the data b The model obtained in the previous step in terms of base e is y Type integers or decimals rounded to three decimal places as needed The population is increasing by approximately 1 cach your e x Number of Years after 1969 1 1970 11 1980 21 1990 31 2000 41 2010 y Populati millions 203 3 226 5 248 7 281 4 308 7

a Let A be a 2x3 matrix B be a 3x2 matrix Show that det BA 0 b Give an example of a 2x3 matrix A and a 3x2 matrix B such that det BA 0

The estimated production of natural gas in a certain country for the years 2013 through 2020 in millions of cubic feet is shown in the table Answer parts a through d below 20000 Year 2013 2014 Av 2015 2016 Natural Gas millions Year of cubic feet 50 110 210 420 y 0 Use integers or decimals for any numbers in the expression Round to the nearest thousandth as needed b Graph the model and the data on the same axes and comment on the fit Choose the correct answer below OA OB O C Round 2017 2018 2019 2020 a Find the equation of the exponential function that models these estimates of the number of millions of cubic feet of natural gas produced in the country y as function of the number of years x after 2010 a Natural Gas millions of cubic feet 600 900 1600 2900 200004 a 20000 Q Q E O D 20000 A c What does the model estimate that the number of millions of cubic feet of natural gas in the country will be in 2025 million cubic feet Round to the nearest whole number as needed d When does the model estimate that the number of millions of cubic feet of natural gas in the country will be 16 000 on the meter

The vectors v Write 11 1 H 5 2 v v 1 H 11 Write 11 0 span R2 but do not form a basis Find two different ways to express COLE as a linear combination of V V V when the coefficient of v is 0 as a linear combination of V V V3 when the coefficient of va is 1 1 as a linear combination of

The total company restaurant expenses incurred by Burger King Holdings Inc between 2004 and 2006 are given in the table Total Company Restaurant Expenses millions 1087 1195 2006 1296 Source Burger King Annual Report 2006 Without using regression the mathematical model for this data is E x lequation Fiscal Year 2004 2005

Consider the subspace of R Create a basis for U 2 U span 0 3 0 4 4 100 3 4 4 2 TOOT 0 0

ind a matrix P such that PTAP orthogonally diagonalizes A Venity that PAP gives the proper diagonal form Enter each matrix in the form row 11 row 21 where each row is a comma eparated list A 6 12 12 0 12 0 12

8 Suppose matrices in PA LU are 0 1 0 0 0 0 1000 0001 001 2 2 1 4 2 9 1 5 3 2 2 1 1 1 4 5 11 O 2 O 100 1 1 1 0 a What is the rank of A b find a basis for the row space of A e True of false rows 1 2 3 of A are linearly independent d Find a basis for the column space of A e What is the dimension of the left nullspace of A e 0 214 2 0 10 1 500 0 0 1 1 3 2 00 0 2 00000

4 Determine if b is a linear combination of a az az 2 a1 a b

1 point Let U U2 U3 be the vectors in R defined by 24 0 8 U 1 U 18 V3 19 c Type the dimension of span U2 U3 15 a Is U U2 U3 linearly independent Write all zeros if it is or if it is linearly dependent write zero as a non trivial not all zero coefficients linear combination of U1 U2 and U3 0 12 U 0 U b Is U2 U3 linearly independent Write all zeros if it is or if it is linearly dependent write zero as a non trivial not all zero coefficients linear combination of U and 03 U2 12 U3 U3

2 Let A 1 2 3 022 The columns of A span R

Find the value of k for which the matrix has rank 2 B 9 A 5 4 2 16 1 9 9 k

Let L R R be the linear mapping defined by Let B L X1 X2 X3 x1 2x2 2x3 x2 2x3 x1 2x2 2 Find the matrix of L EM 13 2 0 16 HD with respect to B and use it to determine L x B and let R such that

te of Use the graph below to find the value of y AS 3 7 4 radians Choose 2 radians 3 2 radians EN 4 radians Choose Choose Choose sin 0 for each value of 0

point Find the rank and the nullity of the matrix ank A ullity A sing your previous answers is the matrix A invertible choose A 2 1 2 1 1 1 7 1 9 3 3 w w b 0 0 3 3

Find the following matrix product if possible 7 2 2 7 OA Select the correct choice below and if necessary fill in the answer box to complete your choice OB The product is not possible 7 2 7 2 2 5 2 2 2 2 5 2

Find the following matrix product if possible 4 6 4 3 5 8 O A Select the correct choice below and if necessary fill in the answer box to complete your cho 4 6 4 2 9 1 3 5 8 2 9 1 31 OB The product is not possible 11

If u and are the vectors below solve the following equation for the vector x 4 7 5 2 1 u 1 x 3u 37 2 2x 0 x 0

B 11 N E a 5 8z 1 4m 7k 5 5 11 the values of the variables 11 6a 7z 12m 5k 5 2 12 74 96 36 10 7

rob 5 5pts Vectors and y in R21 are found such that 10 y 1 and y y 3 a Find 27 3y Show your work

answer true or false and give an explanation 21 If A is 2 x 3 matrix and A 0 9 6 2 and 4 i 1 then 4 6 A 1 A 0 1 Go to page 40 22 If A is a 5 x 3 matrix then null A 03 23 If A is a 3 5 matrix then null A 0

Find the size of the matrix Determine if it is a square column or row matrix 539 595 355 The matrix is X 0 Choose the correct type of matrix O square matrix row matrix Ono special type Ocolumn matrix

Perform the row operation R and replace R on the following matrix 8 800 72 0607 0065 800 72 0607 006 5 0607

Let B 2x 1 3x and B be bases for P and suppose the transition matrix from B to B is 3 a Find PB B PB B

1 3 5 7 associated least squares error Question 5 Let A 1 31 1 and b 2 535 3 Find a least squares solution of Ax b and compute the

Find the values of the variables in the equation 1 X y Z 8 3 3 7 8 x y z

Suppose A 1 mark 1 A 1 1 A A 0 196 0 1 1 020 0 01 0 02 A 1 0 00 1 1 1 1 1 0 1 0 0 1 1 1 10 1 0 0 1 1 0 1 1 0 1 0 20 0 01 0 0 0 2 10 0 02 0 00 2 20 0 0 1 C 02 0 0 10 02 0 00 2 1 011 Which of the following product of matrices are A 001 1 1 1 1 0 1 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 1 0 1 1 0 1 0 1 1 1 1

Question No 2 For which real values of a do the polynomials P t at t P t at pa t t t a form basis for vector space P

A is a 2x2 matrix You are told that 4 3 3 and A 3 2 3 Then the general solution of y Ay is O C e t C te t O C e t C e t t 7 3 3t 3t C e t 3 C e t t 7 O C e t C e 3t t C e t t 2

Find a formula for A using the technique covered in class You may use Maple to comput the eigenvalues and eigenvectors of A You must show everything else A 7 2 4 1

23 For each of the following matrices find an invertible matrix P and a diagonal matrix D such that A PDP a A b A 1 2 3 4 2 i 1 12 2

State if each pair of triangles are similar If by which method 1 2 H S 48 N241 ALMN R 77 84 63 9 A F G H G J 11 12 so state

Let A 5 2 2 1 22 4 a Find all eigenvalues of A over R b For each of the eigenvalues you found under a find the corresponding eige

8 Solve the inequality 5 x x 1 AI

Exercice 3 Let A Compute R A 21 1 3 0 4 1 42 4 4 1 5 5 2 1 3 1 0 4 3 634 8 1 9 5 0 0 3 3 0 0 3 2 14 1 13 3 84 4

0 C Let S 1 mark Which of the following statement is true S is orthogonal but cannot be normalized to an orthonormal set S is not orthogonal S can be normalized to an orthonormal set S can be normalized to an orthonormal set 0 2 2

Exercice 9 Let A 0 1 1 1 0 1 1 1 0 1 10 and B 0 001 000 1 Judge if A could be similar to a diagonal form Give sufficient evidence to support your judgement 2 Find all the eigenvalues of B 3 Judge if B could be similar to a diagonal form Give sufficient evidence to support your judgement

For the matrix A Select one 1 we have 1 True O False AT A 1 9 2

0 0 0 0 0 0 S2 4 O 2 5 72 S3 1 mark Drag the options on the left and drop them into the options on the right S1 52 S3 54 55

Suppose a four sector economy presents the following internal consumption matrix A and external demand vector d A 0 100 0 0 50 0 0 010 0 0 020 0 025 0 0 080 0 0 0 030 0 0 0 020 Vector D 10 20 30 40 A 0 100 0 0 010 0 0 0 030 0 080 0 0 050 0 0 020 0 025 0 0 0 0 020 d 10 20 30 40 Find the production vector to satisfy this demand accurate to four decimal places by using technology 1 Invert the appropriate matrix 2 Invert the appropriate matrix using the power approximation up to power 5 3 After completing parts 1 and 2 discuss your observations

2 Let A 1 2 1 1 1 302 2 0 1 1 3 1 2 5 13 2243 5 a Find N A with it basis b Find the basis for Col A