Matrices & Determinants Questions and Answers

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

A few steps in the process of simplifying the given matrix to row echelon form with 1s down the diagonal from upper left to lower right and Os below the 1s are shown Fill in the missing numbers in the steps that are shown Fill in the missing numbers 1 3 17 4 2 4 3 1 5 1 1 3 17 0 0 2 11 O n D 1 1 3 17 0 2 4 4 3 1 5 11 1 1 3 17 0 2 0 2 1 1 3 17 0 0 1 2

Exercice 1 Let 1 1 Find A4 2 Find A for n 5 A 0 1 0 0 0 00100 00010 0 0 0 0 1 00000

Suppose A and B are square matrices of the same size Which of the following are NOT necessarily true AB A B A B A B A B A B A 2AB B O A B A B A B A AB B O A B A AB BA B AB A 2AB B

4 14 5 Given A 6 14 4 observe that the first column plus twice the third column equals the second column Find a nontrivial solution of Ax 0 without 4 16 6 performing row operations Hint White Ax 0 as a vector equation C

Given the following matrices A and B find an invertible matrix UI such that UA You can resize a matrix when appropriate by clicking and dragging the bottom right corner of the matrix A 3 3 0 4 10 9 9 2 9 9 1 5 9 1 10 9 B 43 33 36 4 10 9 9 2 18 18 2 10 9 1 10 9 oo ol

and the value s of h for which the vectors are linearly dependent Justify your answer 1380 7 The value s of h which makes the vectors linearly dependent is are because this will cause Use a comma to separate answers as needed SELLO to be a varia

13 53 Construct a 2 x 2 matrix C such that AB AC but C B Consider the matrices A C 1 3 O no O yes and B There are many possible correct answers for matrix C You just need to find one of them Moral If AB AC can we conclude that B C 5 3

4 points a 15 points Can B find scalars 2 such that B A A be written as a linear combination of A 6 and A 1 If so

Orthogonal Projection II ind orthogonal projection of the vector 7 X nto the subspace Answer W span 3 0 9 Your response 31 18 5 9 1 18 Correc

A 1 1 0 2 13 1 1 0 2 1 3 2 2 1 0 1 7 2 1 a Find a row reduced echelon matrix R which is row equivalent to A b Consider a homogeneous linear system whose augmented matrix is of the form

The 4x4 matrix uses excel 1 2 3 4 1 10 5 2 11 4 3 15 5 4 20 1 det A 1 4224 solve A 1 4 6 0 3 9 4 5 1 2 3 4 1 0 01183712 0 03811553 0 0234375 0 09446023 2 0 18229167 0 01302083 0 0390625 0 05468750 3 0 17424242 0 01893939 0 1250000 0 17045455 4 0 01089015 0 15506629 0 1015625 0 16690341

3 11 B B 2 0 2 Show whether this list of three vectors is linearly independent or linearly dependent 2 5 points Given a list of three vectors V Hint you can form a matrix V V V3 then use the reduced echelon form It also works if you use the definition of linear in dependence

Find the geometric and algebraic multiplicity of each eigenvalue and determine whether A is diagonalizable If so find a matrix P that diagonalizes A and determine P AP Notice that the order of the eigenvalues and corresponding eigenvectors can be different from yours and that the eigenvectors are defined accurately to the factor sign 2 0 0 0 A 0 2 50 50 0 3 0 00 3 0 0 O 2 Algebraic multiplicity Geometric multiplicity 2 3 Algebraic multiplicity Geometric multiplicity 2 2000 02 0 0 00 3 0 Loo 0 3 O 2 Algebraic multiplicity Geometric multiplicity 2 3 Algebraic multiplicity Geometric multiplicity 2 2 0 0 0 0 200 0 030 003 0 P P 0100 10 10 10 00 0 1 00 1 0 01 0 0 10 50 50 00 0 1 00 1 P P AP OX 2 Algebraic multiplicity Geometric multiplicity 1 3 Algebraic multiplicity 2 Geometric multiplicity 1 A is not diagonalizable P AP OX 2 Algebraic multiplicity Geometric multiplicity 2 X 3 Algebraic multiplicity Geometric multiplicity 2 01 0 0 10 10 10 00 0 1 00 1 0 PAP 2 0 0 0 0 200 0 030 0 0 0 3 O 2 Algebraic multiplicity 2 Geometric multiplicity 1 A 3 Algebraic multiplicity Geometric multiplicity 2 A is not diagonalizable

Use the method of Example 6 to compute the matrix A 0 where 1 2 A 1 2 2 i 2048

Recall that R m n denotes the set of all m x n matrices with real number entries How many of the following assertions are correct 2 4p Recall that Rm denotes the set of all m x n matrices with real number entries How many of the following assertions are correct i In a product of matrices AB the matrix B maps rows of A to rows of AB ii In a product of matrices AB rows of A are multiplied by columns of B iii A matrix A of size m x n maps n columns x to m columns Ax

3 Find the matrix of the transformations T R2 R2 in the basis B 2 a T is reflection with respect to the line ya b T is projection onto the x axis for each of the following transformations

A 2 1 1 1 Does A has a one sided inverse If yes find at least two different ones 1

Let u and v be two vectors such that If u k 1 1 1 then k 12 48 12 144 4 u 12 v 3 compvu 4 and projuv

Tute eigenvalues and bases for the eigenspaces of AP for O A 1024 A 1024 GIGIFI CAH O A 1024 0 4 O A 1024 A 4 EG A 4 DE 0 Sheet 4 LA 8 8 4 0 8 4

1 estion 1 6 points Let A 2 i i 2 i 7 4i 28 i 10 12i a Show that A is positive definite by using principal submatrices of A b Find the Cholesky factorization or Cholesky decomposition of A c Solve the system Ax b 2 9 3 3 12 b

Question 2 4 points Find a QR factorization of A 2 2

2 2 3 A 2 32 4 25 Eigen value and basis basis for eigen space by A 3 1 1 A 10 1 1 1 E All False 2 B 2 20 1 1 00 01 2 C 30 4 23 1 1 2 1 1 23 5 1 H 1 1

Problem Let A 8 7 points 11 1 2 22 3 1 1 10 5 A 2 G G 6 0 G C 3 D 2 5 G B 1 2 Find a basis for the column space of A

Find a basis for S AE M x2 the sum of the entries of A is f A is 0

Problem 5 7 points Find the coordinate vector v where v 3 x 62 and 8 1 2 2 2 2x x A 3 1 6 B 1 3 6 7 3 2 D 2 7 6

2 Let M be the magic 6 x 6 matrix Determine the matrix U2 I where U is the upper triangular matrix in the LU decomposition of M

5 Use the table below to predict how long it would take to walk 40 6 miles Round your answer to the nearest hundredth Time Miles 0 4 95 9 72 15 75 18 99 27 27 32 37 0 1 65 3 24 5 25 6 33 9 09 10 79 O 14 57 hours O 13 78 hours O 13 52 hours 14 3 hours

Simple linear regression Explained and unexplained variation Bivariate data obtained for the paired variables x and y are shown below in the table labeled Sample data These data are plotted in the scatter plot in Figure 1 which also displays the least squares regression line for the data The equation for this line is y 253 74 0 96x a In the Calculations table are calculations involving the observed y values the mean y of these values and the values y predicted from the regression equation Sample data X 111 2 154 1 117 2 134 1 127 9 130 6 137 6 115 2 152 2 111 6 Send data to Excel Answer the following Calculations 5 624 0004 24 8004 2 1904 193 7664 306 9504 Column sum 1151 7080 50 5805 50 8084 0 1267 41 5251 15 7768 Column sum 158 8176 319 2654 146 6037 3 3709 55 8906 461 9061 Column sum 987 0366 100 IM in 13 130 DR Tin 13 EM 140 190 Figure 1 a The variation in the sample y values that is explained by the estimated linear relationship between x and y is given by the Choose one which for these data is Choose one b The value is the proportion of the total variation in the sample y values that is explained by the estimated linear 2 relationship between x and y For these data the value of is Round your answer to at least 2 decimal places c The least squares regression line given above is said to be a line that best fits the sample data The term best Att X G Espa ol

the system of equations corresponding to the augmented matrix Then perform the row operations R 4r2 and R3 2r2 r3 on the given augmented matrix 9 4 1 5 2 4 7 5 14 2 4 2x 4y 7z 5w 4x y 4z 2w 9x 4y z 5 2x 4y 7z 5 4x y 4z 2 O D 2x 4y 7 5 4x y 4 2 9x 4y z 5 0 2x 4y 7z 5 0 4x y 4z 2 0 Perform the row operations R 4r2 r and R3 2r2 r3 on the original matrix given in the problem statement and enter the resulting matrix below 0000

Suppose A has eigenvalues 3 1 with corresponding eigenvectors u U EASANT Find the following Express your answer in terms of us and 2 enter u as u 1 and u as u 2 A 2u A 5u A 31 2 100 20

In Exercises 13 14 find proj u 13 a u 1 2 a 4 3 b u 3 0 4 a 2 3 3 14 a u 5 6 a 2 1 navig b u 3 2 6 a 1 2 7

y 2x 14 are O 5 24 4 22 O 8 30 4 6 O 5 4 4 22 O 8 24 4 6 Write and solve the equation based on the following dia X 1 1 1 X X I 1

Question 1 Evaluate the determinant of the matrix using theorem 3 6 40 25 10 A 30 5 20 15 35 45

A pine tree growing on a hillside makes a 77 angle with the hill From a point 82 feet up the hill the angle of elevation to the top of the tree is 64 and the angle of depression to the bottom is 19 Find to the nearest tenth of a foot the height of the tree

4 If x 5 y 3 and z 12 calculate the following EZ A

12 Let x 1 y 4 and z 4 Evaluate 4x 6 1z 2y

Use row reduction to evaluate the determinant of the matrix Do not use cofactors 3 0 2 6 9 0 2 1 5

Solve the given system of differential equations by systematic elimination Dx y Dy z Dz x x t y t z t

Find the volume between the cone x y z and the sphere x y z 25

7 Is matrix A 2 diagonalizable A If yes diagonalize it then find A

9 What is the cosine of angle between matrices A and B with respect to standard inner product on M22 1 31 A 2 21 B 23 1

10 Is S V1 V2 V33 where v 2 1 0 v 1 2 0 and v 0 0 1 an orthogonal set with respect to the Euclidean inner product on R If yes find the corresponding orthonormal set

6 Find the bases for the row space column space and null space of the following matrix then find A 4 51 A 3 2 2 1 10 8

The profit function P is shown for producing and selling x items Determine the values of x for which a P x 0 the company breaks even b P x 0 the company experiences a loss c P x 0 the company makes a profit POO 760 y P x

Suppose the production of 1 unit of mining requires 2 5 unit of mining 1 10 unit of manufacturing and 1 2 unit of communication To produce 1 unit of manufacturing requires 1 10 of mining 3 10 of manufacturing and 3 5 of communication To produce 1 unit of communication requires 1 10 units of mining 4 5 units of manufacturing and 1 10 units of communication a 3 points Set up a 3 by 3 matrix for the input output model b 6 points Find the ratio of the three commodities in the closed model To receive full credit the work must be submitted

C x 35 8x 3776 Part 2 of 4 b Write a linear revenue function that represents the revenue R x for selling x items The linear revenue function is R x 47 6x Part 2 4 Part 3 of 4 c Write a linear profit function that represents the profit P x for producing and selling x items The linear profit function is P x

Compute the product AB by the definition of the product of matrices where Ab and Ab are computed separately and by the row column rule for computing AB 1 3 3 4 5 4 A B Set up the product Ab where b is the first column of B Ab Use one answer box for A and use the other answer box for b Calculate Ab where b is the first column of B Ab Type an integer or decimal for each matrix element Set up the product Ab where b is the second column of B Ab Use one answer box for A and use the other answer box for b Calculate Ab where by is the second column of B Ab Type an integer or decimal for each matrix element D COLOD for the first entry in the first column of AB using the row column rule Choose the correct answer below

Find the values of x and y using pseudoinverse X 3y 5x 7y 11x 13y 17 19 23 P aa Ta P A ATA 1AT A

4 In the triangle below 5 represents which ratio A 3 C O sinc OsinB OcosC tanB 4 5 B