Matrices & Determinants Questions and Answers

Explaining Why a Set Is Not a Basis In Exe 15 22 explain why S is not a basis for R 15 S 1 3 0 4 1 2 2 5 2 16 S 2 1 2 2 1 2 4 2 4 17 S 7 0 3 8 4 1

Perform the elementary row operation 14 2 35115 53 5 Resulting matrix R3 R3 R3 on the given matrix Write numbers as integers or simplified fractions 4

Perform the elementary row operation 3R R on the given matrix 1 4 9 4 3 0 6 i Resulting matrix 0 X S

y n 1 to Tn 1 0 satisfying the Given any initial values y to ro there always exists a unique solution to y n t a y n 1 t any t given initial values

nd the matrix A for T relative to the basis B T R 1 11 3 1 2 2 R T x y z x y 4z 4x y z x 4y z B 1 0 1 0 2 2 1 2 0 2 3 4 3 4 3 5 3 13 3 4 3

3 If T R R is a linear transformation the vectors e e2 and e3 span determine a unit cube of volume 1 cubic unit and the vectors T e T e and T e3 span a parallelepiped of volume det M where M is th matrix of the transformation T 0 If the vectors 5 3 and 2 are the transformed basis vectors 62 then the volume of the parallelepiped spanned by these vectors is

Verify that X t is a fundamental matrix for the given system and compute X t Then use the result that if X t is a fundamental matrix for the system x Ax then x t X t x 0 x is the solution to the in alue problem x Ax x 0 xo 060 x 1 0 1 X 1 1 0 x 0 X Ax X AX2 X3 AX3 2 0 5 X t 24e 6e 2t 4e t 20e t 6e t 2e 2t 3e t 2e 2t 3e t C 060 a If X t x t x t X3 t and A 1 0 1 validate the following identities and write the column vector that equals each side of the equation 110 b Next compute the Wronskian of X t W X1 t x t X3 t Since the Wronskian is never r and each column of X t is a solution to x Ax X t is a fundamental matrix

1 For the following linear programming problem Minimize z 4x 6y subject to x 3y 25 2x y 24 20 y20 i Solve it using any method not forgetting to find the optimal value ii Set up the dual problem iii Solve the dual problem using any method not forgetting to find the optimal value iv If you solved them both correctly the optimal values for i and iii are the same Is t 2 If not what theorem done it follow from

1 2 FHI Let u 1 2 4 1 2 2 U u u W be the column space 0 of U and w be the orthogonal projection of y onto W and z y w Determine which of the following is false 1 2 0 1 3 a w UUT y 4 24 b w 5 c z 3 d The minimum distance between any v EW and y is z e none of these

Solve the system of linear equations and check any solution algebraically If there is no solution enter NO SOLUTION If the system is dependent set a and solve for x and y in terms of a x y z 2x 4y Z 1 X 2y 3z 3 x y Z 2

s a b of GL 2 R such that ab bc 1 Show that H is a Let H be the set of elements subgroup of GL 2 IR b How many subgroups does Z36 have What are they

Find the eigenvalues and eigenvectors of the linear transformation T P P given by T ax b 6a 10b x 2a 3b Eigenvalue 2 Eigenvalue A Ex 59 Eigenvector 5x Eigenvector x 1

Find the plane of reflection of the following pure reflection map Q 3 ON ON AIN 7 7 6 6 2 2 191201 7 21 37 67 7

Use the Gram Schmidt orthonormalization process to transform the given basis for R4 into an orthonormal basis Use the Euclidean inner product for R4 and use the vectors in the order in which they are given 3 4 0 0 1 1 0 0 2 1 0 1 0 1 1 0

e graph shows the number of words in millions in a particular government document for selected years from 1950 through 2000 The data can be modeled by f x 0 15x 1 24 and 1 790 0368x in which f x and g x represent the number of words in millions in the document x years after 1950 Use these functions to complete parts a through c below According to the linear model how many millions of words were in the government document in 1990 There were approximately million words in the govemment document in 1990 Do not round until the final answer Then round to one decimal place as needed 16 10 18 512 26 41 5 43 Gulmon G 1950 1960 19707900 19902000

et A 8 0 6 24 12 4 0 12 10 4 points Find the eigenvalues of A 6 points For each eigenvalue A of A find a basis for the eigenspace of X 6 points Is the matrix A diagonalizable If so find matrices D and P such that D is a diagonal matrix and A PDP Eplain carefully why not

Let A A a Determine the eigenvalues A and X2 of A where X d X 5 0 4 6 3 12 201 m x b Determine the algebraic multiplicity m of each of the eigenvalues in part a m x

60 DE NA 60 2 80 407 80 B 2 5 C Q i ii B L 2 7 M2 PE iii 2 5 80P 4 B D M 99 F 40 R M 2 5 70 N 80 F 5 Q4 iv 70 ESP 10 Q 86 30 R R R

3 Consider the matrix 2 a Find elementary matrices E and E such that E E A I b Use part a to write A as a product of elementary matrices

tan f x 1 x 2 Answer tan x lx x 2 4 then find the value of x

f x x 4x x x 4 shown below D The graph as to the land ses to the hymn OC The graph rises to the left and falls to the right COD The graph rises to the left and to the right b Find the x intercepts State whether the graph crosses the x axis or touches the x axis and turns around at each intercept What are the x intercepts x 4 1 1 Type an integer or a decimal Use a comma to separate answers as needed CV At which x intercept s does the graph cross the x axis Select the correct choice below and if necessary fill in the answer box to complete your choice A x 4 1 1 Type an integer or a decimal Use a comma to separate answers as needed OB There are no x intercepts at which the graph crosses the x axis At which x intercept s does the graph touch the x axis and turn around Select the correct choice below and if necessary fill in the answer box to complete your choice A x Type an integer or a decimal Use a comma to separate answers as needed B There are no x intercepts at which the graph touches the x axis and turns around c Find the y intercept The y intercept is y 4 Type an integer or a decimal d Determine whether the graph has y axis symmetry origin symmetry or neither Choose the correct answer below O A y axis symmetry 10 8 6 2 6 2 8 10 101 12 8 6 Ay X 8 10

Find the QR decomposition of the matrix M 1 25 7 2 3 21 1 3 1 4 1 2 3 25

By using properties of determinants show that 10 a a 0 c 0

4 A survey of the ages of 90 retirees are organized in the table below V Veteran Veteran Non veteran TOTAL N Non veteran p E70 70 79 25 23 48 80 89 12 19 31 E70 age 70 to 79 90 99 4 7 11 Eso age 80 to 89 n Eso TOTAL 41 49 90 Ego age 90 to S p N n E90 n V U EBO p N E90 p E70 V Are the events Being a veteran and Being age 70 to 79 independent or dependent Prove mathematically

2 Consider the following model a identify matrices c A and b Max 2500x 1600x s t X 210 X 2 15 5x 9x 320 1x 2x 2 300 12x 15x 2 480 x 0 x 20 b Solve the problem using Simplex Algorithm

Given matrices A and B shown below find 8 A 4 B Give your answer as a matrix using the matrix tool Provide your answer below A B 3 0 71 4

3 Find the standard matrix of the transformation T defined by 2x 3y z W 5x 2y z W2 3y 5z W3

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients i x 2x 8 ii 4s 4s 1 iii 6x 3 7x iv 4u 8u v r 15 vi 3x x 4

Ex 5 R 5x1 x2 X3 x 2x3 2 ED T 3 1 Ex 5 R T X1 x2 Ex 5 Ex 5

Solve the system in terms of the arbitrary variable x 3x 6y z 7 x y 3z 5 The solution set is 00 Simplify your answer Type an expression using x as the variable Use integers or in the expression

Subtract the following matrices 4 0 2 3 Give your answer as a matrix using the matrix tool 31 11 3 4 3

Let A 3 1 0 2 0 2 4 a 4 points Find the eigenvalues of A b 8 points Find the basis eigenvector s for the smallest eigenvalue

Solve the system State whether the system is inconsistent or has infinitely many solutions If the system has infinitely many solutions write the solution set with y arbitrary 5x 6y 5 10x 12y 10 Select the correct choice below and if necessary fill in the answer box to complete your choice O A There is one solution The solution set is Simplify your answer Type an ordered pair OB The system has infinitely many solutions The solution set is Simplify your answer Type an ordered pair Type an expression using y as the variable C The solution is inconsistent B

1 5 pts Let L R R2 be the linear transformation defined by L x a Find a basis for L S where S Span 2 0 1 0 1 1 21 22 21 2x3

Define T R R by T x A where matrix A Row1 4 3 Row2 4 3 Row3 4 3 What is the image of u 4 1 under this transformation

Let A 1 3 1 Find an orthonormal basis of the column space of A 1 3 3 6 1 3 27 3 4 6

8 3 9 8 1 9 A B 6 9 7 8 2 AB AC A Ex 42 C

1 Two line segments AB and AC include an angle of 60 where AB 5 cm and AC 7 cm 3 1 Locate points P and Q on AB and AC respectively such that AP AB and AQ AC 4 Join P and Q and measure the length PQ Sol i Draw LBAC 60 such that AB 5 cm and AC 7 cm X X X A 27 60 7 cm Q3 3 cm 5 cm P B

4 Construct a pair of tangents to a circle of radius 4 cm from a point which is at a distance of 6 cm from the centre of circle Sol The distance of point from which tangents to be drawn should be more than radius so that tangents can be drawn P M 4 cm FX

14 From the top of a tower n m high the angles of depression of two objects which are in line with the foot of the tower are a and a Find the distance between the two objects Sol Consider a vertical tower TW h m Two objects A and B are x m apart in the line joining A B and W and BW y let The angle of depression from top a tower to objects A and B are a and respectively T In ATWB we have h

How Do You See It 38 The graph shows the total revenue and expenses of a company x years after it opens for business a Estimate the point of intersection of the graphs b Interpret your answer in part a Millions of dollars 4 6 2 0 0 expenses 2 4 revenue 6 Year 8 10 X

Let A D PT AP 23 8 8 8 11 16 Find an orthogonal matrix P with rational entries and a diagonal matrix D such that 8 16 11 P D

Uni tanily diagonalize the quadratic Form Q 21 2 2 23 4x 4x 4x 2x x2 221x3 2222 You have to show work clearly

Calculate by finding Jordan canonical form A 2

4 Show that if A is symmetric then so is PT AP for any compatible choice of F 5 Generalize Exercise 2 to the case of complex matrices

find the singular value decomposition of each of the following matrices a b 1 Hi 1 12 2 1

a b 3 a b 8 c d a be a basis for R x2 and 1 2 be a basis for P Let T R2x2 P be a linear transformation defined by T Let B Find the standard matrix of T Ex 5 Ex 5 C B Fv 5

If A and B are similar matrices then both of them are diagonalizable True or False explain

Solve the following system 2y z 7 x 4y 14 3x 2z 8 x 8

Find the characteristic equation and the eigenvalues and a basis for each of the corresponding eigenspaces c a the characteristic equation b the eigenvalues Enter your answers from smallest to largest d 2 a basis for each of the corresponding eigenspaces