Matrices & Determinants Questions and Answers

Select a spanning set for the subspace S x y z ER 3x 2y z 0 of R 1 1 0 2 1 1 3 2 1 0 0 2 1 O 1 1 1 0 1 2

Find the values of for which the determinant is zero Enter your answers as a comma separated list 2 4 0 5 3 OO 24 2

1 Use expansion by cofactors to determinant of the matrix 0 43003 1 6 2 2

15 Subject to which constraint the function f x y z x y 2 does NOT attain its maximum value A x y z 1 B x y z 1 C ez y 1 D er 2 x z y

89 Basis and Dimension Find a basis for and the dimension of the subspace of R spanned by the following sets a 0 1 2 3 0 1 3 2 3 n 3 b 1 3 1 2 7 6 n 2 c 1 2 0 4 3 1 1 2 5 3 1 6 7 0 2 0 n 4

7 marks Let P be the plane through the point A 2 1 1 with normal vector n 2 2 1 a Compute parametric equations of the plane P b Compute the distance from P to the origin

3 Consider the subspace W span V V2 V3 Spon C100 10 3 1 1 1 1 20 0 Let v 1 2 4 1 i Find the orthogonal complement of W ii Find a basis for the orthogonal complement of W ili Find the projection of v on the orthogonal complement of W

In each part the augmented matrix for a linear system is given in which the asterisk represents an unspecified real number Determine whether the system is consistent and if so whether the solution is unique Answer inconclusive if there is not enough information to make a decision O Inconclusive O The system is inconsistent it has no solution O The system is consistent it has a unique solution The system is consistent it has infinitely many solutions

1 If A M16 8 IR then a The minimum Rank would be b The maximum Rank would be c The minimum Nullity would be d The maximum Nullity would be 2 If A M5 14 IR then a The minimum Rank would be b The maximum Rank would be c The minimum Nullity would be d The maximum Nullity would be 3 If A M11 11 IR then a The minimum Rank would be b The maximum Rank would be c The minimum Nullity would be d The maximum Nullity would be

5 Let T R T x 1 R2 be a linear transformation defined by 1x Find the kernel of T

Let 183 X1 X2 8 X3 883 Describe all solutions of 31x1 x 3x3 Z4X4 4 18 x2 1 4 3 z 7 21 23 z 3 x4 04 9 27 000

What is the y intercept of the line shown below The y intercept b is the y value when the graph crosses the y axis Here the graph crosses at the point 0 2 making b 2 A b 2 y 5 5 5 10

4 For given equations 10x 2x2x3 27 3x1 5x2 2x3 61 5 x x2 6x3 21 5 a Solve by naive Gauss elimination b Solve by LU factorization Solve by iterative method

2 For each triangle use two different methods to determine x to the nearest tenth of a unit a b 35 X 1 0 cm 0 7 cm X 24 8 8 km 3 6 km 6p

Suppose T R R4 is a linear transformation with T standard matrix A such that T x NOTE e refers to the ith column of the n x n identity matrix Ax 17 9 3 T e 14 8 12 Find the

35 Myranda received a movie gift card for 100 to her local theater Matinee tickets cost 750 and evening tickets cost 12 50 each the possible If x represents the number of matinee tickets she could purchase and y represents t of evening tickets she could purchase write an inequality that represents all the p Myranda could spend her gift card on movies at the theater On the set of axes below graph this inequality Number of Evening Tickets Number of Matinee Tickets X What is the maximum number of matinee tickets Myranda could purchase with her gift card Explain your answer 36

Which of the following are true for both absolute value and square root parent functions A II and III B L II and III C II III and IV D I and III I THI IV Domain of All Real Numbers Range y 20 Contain the point 0 0 Contain the point 3 3

would your formula look like Use the mock spreadsheet below as a reference 5 C 0 00 0 12 Hint 1 Day 2 Yesterday 18 3 Today 30 4 A new old old A B T B Closing Amount in cell C3 to find the percent change in the B column What 82 83 B2 82 83 C D Speed 1x 1

11 In 1990 the city of Andover had a population of 15 000 people Since then the population has grown by 3 3 each year In 1990 the city of Minneapolis had a population of 370 000 and has been decreasing by 1 7 each year How many more people will be living in Minneapolis compared to Andover after 25 years A 207 237 B 234 530 C 530 155 D Because Andover is increasing and Minneapolis is decreasing Andover will have

Match the correct definition with the type of income consistent Income Inconsistent Income a Your paycheck is a set amount matter the hours you work b Your paycheck will change based o the hours your work

Question 1 The table of values represents a linear function g x where x is the number of months that have passed and g x is the value of the vehicle X 0 4 8 Show Transcribed Text A y 20 500 1000 x 4 B y 20 500 1000 x 4 g x 21 500 20 500 19 500 10 500 3506 41 G stion 1 4 tion 1 Part B The table of values represents a linear function gix where x is the number of months that have passed and g x is value of the vehicle e the equation of the line in point stope form 1 point U Show all work 3 points

Consider the matrix A below 09 A 3 5 6 Determine the inverse of A and enter your matrix entries in cells below A 1

Score for Question 2 of 4 points 4 2 v 9 2 is transformed using the transformation matrix A 1 What is the resulting vector Show all your work Answer 2

2 Is b a linear combination of a and a2 Let 0 3 3 and b 2 Yes b is a linear combination of a and a2 Ob is not a linaer combination of a and 2 We cannot tell if b is a linear combination of and 2 Either fill in the coefficients of the vector equation or enter DNE if no solution is

3 Is a linear combination of a and a Let B 1 2 1 Yes 3 and Ob is not a linaer combination of a and We cannot tell if is a linear combination of a and bis 0 A 2 a linear combination of a and Either fill in the coefficients of the vector equation or enter DNE if no solution is possib a az

3 State the type of system of equations y 4x 1 20x 5y 5 O Inconsistent Consistent and dependent Consistent and independent O None of the above

Problem 5 3 points Consider the square in R with corners at 1 1 1 1 1 1 and 1 1 There are eight symmetries of the square in cluding four reflections three rotations and one identity symmetry Write down the matrix associated to each of these symmetries with respect to the standard basis

Solve the following eigenvalue problem AX 2X 1 1 1 11 A 1 1 1

Let T R R be a linear transformation defined by T v Av 1 0 0 10 1 Find T 3 1 1 L2 1 2 A 2 Mar

Here is the graph of y f x 5 4 3 2 3 Below is a graph of a transformation of y f x 54 3 1 1 2 0 sart x 2 3 Which of these is the correct formula for the transformed graph sqrt is shorthand for the square root of whatever is in the parentheses see How Do I Write Math On A Computer for m details O sqrt x O sqrt x O 1 sqrt x

3 Use Gauss Jordan elimination to solve where possible the systems of equations AZ B with the following augmented matrices giving the complete general solution where it exists If the solution is not unique write it in the form Z Z Zo where Z is a particular solution and Zo is a general vector in the null space 2 1 14 a b 3 c 1 i 2 d 1 i 22 2 1 3 i

3 In each case show that S is a subring of R a b a S a b c d R a c b d R M R b a be R R M R b S a a 0 b 0 C a b c d e R R M3 R c S 00 a d S a b a be R R M R

2 4 6 8 Let A 1 3 0 5 1 1 6 31 a Find the basis for the null space of A b Find the basis for the row space of A c Find the basis for the range of A that consists of column vector

21 197 23 2 OFI 8 8 8 P 6 9 HANS k 1 19 0

18 A ship leaves a Port and heads directly East at 25 miles per hour for 2 hours Then the Captain sees an Oil Rig that is N60 W from its location If he travels for 30 miles towards this Oil Rig how far is the ship from the Port Round to the nearest tenth of a mile

7 Find the area and perimeter of a regular nonagon with a side length of 10 cm Derimoter 8 Find the area and circumference of the circ answers in exact form in terms of Area 12 m Circumference

Consider the matrices 3 B Verify the following expressions where possible and give reasons 5 1 A BC A B C 5 2 AB C A BC 5 3 a b C aC bC where a 3 b 2 5 4 a B C aB aC where a 3 b 2 5 5 A A and A B At Bt A 3 1 2 1 2 1 1 10 1 1 1 0 1 2 2 2 1 0 1 1 1 1 0 1 1 1 0

Which of the following sets is orthogonal ON 4 2 0 0 0 1 1 2 0 OA OB 4 3 1 0 1 1 1 1 1 1 3 0 0 0 1 1 1 0 3 3 2 OD 1 2 3 2 4 1

Determine the Magnitude and direction pixel locations 1 3 and 3 4 the following matrix Comment on the results 128 128 128 128 128 128 1 1 1 1 1 128 128 128 128 128 128 128 128 128 128 128 12 8 1 1 1 1 1 1 1 1 1 1 1 1

Check if the given set of polynomials form a basis for P set of all polynomials of degree less than or equal to 2 1 3x 2x 1 x 4x 1 7x

The matrix 6 3 6 3 0 6 has a singular value decomposition A UZVT where and and A U V 15 3 3 6 Sual al sl si 12 0 0 0 00 The 2 1 entry of U is U 21 1 y What is y 6 0 WING 0 1 2 1 xx 2

The matrix 3 3 3 5 5 has a singular value decomposition A U EVT where 0 19 and and A U V 0 5 2 0 3 2 0 1 2 1 2 2 The 2 2 entry of the pseudoinverse A is A 22 1 z What is z

Let B b1 b and C C1 C2 be two bases for R where 3 13 2 0 9 18 C1 b If the change of coordinates matrix from C to Bis C2 025 0 36 026 O 35 O 34 22 2 C what is the value of b

Consider the linear transformation T M22 P2 given by T a b a b c 5d a b c 5d z a b c 5d z Then the dimension of the kernel of Tis A and the dimension of the range of Tis

Let 0 2 0 Find the orthogonal projection of y to W W span 1 2 0 and y

The LU factorization of A A To solve the system 3 1 1 1 5 2 x 0 Ly 1 1 2 Ux 00 1 3 10 0 2 1 1 0 0 7 3 and then solve 16 2 8 28 using the LU factorization you would first solve 9 9 16 91 28 1 20 3 1 5 2 8 1 1 3 2 is 0

3 mi Answer 3 mi Image Description What is the perimeter of the shaded region Give your answer as a decimal number with units What is the area of the shaded region Give your answer as a decimal number with units

Identify the two statements that contradict each other 9 a APQR is equilateral b c APQR is isosceles APQR is a right triangle

Given A 1 3 1 1 2 3 4 6 2 15 and b 1 3 6 a Find the LU factorization of A b Using LU factorization find the complete solution to Ax

1 Find a basis for the subspace of matrices A R 2 that satisfy the following equation 1 4 42 688 1 A 00