Matrices & Determinants Questions and Answers

Let M be the following matrix with entries from Z5 1 1 3 0 2 3 0 1 Which one of the following is a basis for the null space M E E 0 8 8 B TAAL O O 14 20 M

10 Let P 0 001 2 3 0 1 3 and X 1 a Show that XP X Is the result in part a to show that X P 1 0 where 0 is the

Find the real matrix e where O O O 64 C e4 eA 12 O e 2 4 3 1 3 1 2 1 3 1 4 1 10 0 1 e 41 4 12 A

I need the MATLAB codes for TASK 3 only t Task1 Write the symbolic math expression for one period 1 of the periodic signal x t shown in Figure 4 1 xp 1 0 5 0 5 1 1 0 75 0 5 0 25 0 t Figure 4 1 Periodic signal x t 0 25 0 5 0 75 1 Task 2 Write the symbolic Matlab expression corresponding to three periods of x t Use the variable xp to represent the said expression Task 3 Determine the exponential fourier series coefficient Cn Plot the magnitude and phase of Cn for 10 n 10

Consider the real vector space with ordered bases 04 1 id respectively Find the change of basis matrix id O id 11 id 83 id id id 112 112 1 12 12 12112 7 21 2 122 112 V e e 1 21 2 1 B e e and C cosh x sinh x

Consider the following vectors LOT Which of the following statements are true select all correct answers u Vector x lies in the plane spanned by v and w Vector w lies in the plane spanned by u and v Vector x lies in the plane spanned by u and v Vectors u v and w span a plane Vector w is not a linear combination of u and v W X 3 LOT

A QR factorization of A is given Use it to find a least squares solution of Ax b A 21 20 Q N M2 3 3 T W 32 32 H R 5 2

Compute A A and A CO A F A A 00 11 A 0 6 4 5 1 5 3 3 5

stando 5 If A is a matrix that satisfies the equation 2 t t AY 2 A 5A 4 I 0 show that A is invertible and find a formula for A 176 171

4 Solve the linear system 2x2 using Gauss Elimination OR Gauss Jordan Elimination 32 4y 10 7a 9y 20

For the matrices A and B given in Question 3 find BA if possible Not possible 8 16 7 9 3 5 4 8 3 3 1 OT 6 14 7 12 3 5 A IN FORT matices a AD 11 POSSIDIC 2 2 13 24

Let P Find P DP Select one O a O b O C 2 5 a 0 and D 1 3 b O d e 6a5b 26 2an 6a5b 26 2an an 0 06 6a56 26 2an 6a 56 26 2a 15a 156 66 5an 15a 156 66 5a 15 156 66 5an 15a 156 6b 5a where a b 0

For the given graph 6 4 3 2 a The slope m 16 5 b The y intercept b 4 3 2 1 my 2 3 A 5 6 1 2 3 5 6 8 c The equation of the line in slope intercept form i e y mx b is

Op For the following 2D system a find fixed points b linearize the system c classify genvalues of each fixed points and d sketch phase portrait y x x y

2 Row reduce A to reduced row echelon form RREF 1 1 2 3 3 5 3 1 3 5 7 3579 5791 2 2 0 2 a A b A c A 2 3 2 3 2 2 2 0 1 3 3 2 3 2 2 2

cos 240 sin 330 5 sin 6 tan 315

8 Find the diameter of this circle 7 4 2 1 a C

3 2 pt Find the kernel of the linear transformation L R R with matrix 25 1 39 0 14 1

2 points The set sa basis of the space of upper triangular 2 x 2 matrices 33 0 ind the coordinates of M MB with respect to this basis B C 21 69 0

20 points Let 3 10 1 3 3 4 2 and let W the subspace of R4 spanned by u and Find a basis of W the orthogonal complement of W in R

Translate the following figure using vector 2 1 Choose the coordinates of the vertices of the image 4 3 2 0 3 2 1 3 2 3 0 2 3 1 5 4 3 No 2 3 4 16 y H

20 points Let I be the line given by the span of is A basis for Lis 74 6 in R Find a basis for the orthogonal complement L of L 5

Q2 Let W be a subspace of R4 Suppose W has basis a Find an orthonormal basis for W b Is the vector in w Why 0 0

A graph of its boundary lines is shown below Choose the letter of the region which represents the solutions to the system of inequalities given below y 5 y x 3 D A 8 6 4 2 8 y 6 4 2 O 2 4 6 2 4 C 6 B X 8

Let 7 R R be a linear transformation for which Find 13 8 and r x 1 13 4a b 26 3a b xx nd

20 points Let I be the line given by the span of Abasis for Lis 1 1 1 in R Find a basis for the orthogonal complement L of L

Find the coordinate vector xla of x relative to the given basis B b b b b 4 xlo 3 1b 4b3 12 2 6 8 CITS

Question 1 Solve the following linear system using Gauss Jordan eliminatio method 6 points 2x1x2 x3 2 x1 2x2 x3 3 3x1 2x2 2x3 2

Find bases for the null space and row space of A A 1 3 1 0 2 2 4 4 V Enter the basis for the null space and the row space of the matrix A Null space of Ais i 6 5 8 2 1 8 2 3 0 6 4 4 V2 i r 1 Row space of A is Mi T MI 7 5 1 i MI MI IN IN

Solve the following wong maximigre Subject to LP using m method 2 x 502 3x 42 b x 3x2x2 72 O

Let A P 19 4 8 2 3 1 3 2 3 25 4 8 4 Find an orthogonal matrix P with rational entries and a diagonal matrix D such that D PT AP 19 sqrt 5 2 sqrt 5 4 sqrt 5 2 sqrt 5 sqrt 5 3 D 9 0 27 0 27

1 k For what value s of k will A 2 0 3 1 NIT k 0

5 Find the exact value of each expression without using a calculator a tan con 3 COS b sin 105 c cos 7 1 2 COS sin 7 sin 20 2

man is 6 feet tall and casts a 4 foot shadow At the same time a building casts a 24 foot shadow How tall is the building 96 feet O 1 foot O 36 feet O26 feet None of the choices are correct

Solve the system Az b by factorizing the matrix A where 4 1 1 0 1 1 1 4 1 4 1 4 0 1 1 0 1 1 4 A b 2 0

Solve the matrix equation for X X IN x 1 3 19 01 14 2 9 5

The two way table shows the results of a survey that asked students whether they like hip hop music or pop musi Make a two way table that shows the joint and marginal relative frequencies Hip Hop Hip Hop Like 33 Dislike 33 Like Dislike Like Dislike Total Pop 30 54 Pop Like Dislike Total

a Carry out by hand one complete iteration of the rank one method with x 0 0 0 and H 0 I to find an approximation to a local minimizer of f Perform the line searches analytically and find both x and H b How is the rank one method an improvement on the Newton Raphson method for finding a local minimizer State one drawback of the rank one method

Let F2 t denote the field of rational functions in t over F2 a Prove that F t F2 t is not Galois b Prove that F t F t is Galois c For which values of n is F2n t F2 t Galois Justify your answer

Suppose A is the matrix for T R R relative to the standard basis Find the diagonal matrix A for T relative to the basis B A 0 1 A 201 1 0 0 0 1 B 1 1 0 2 1 0 0 0 1

The protection given by the law of the state refers to the service contracts warranties and service contracts O warranties

The term fiscal policy refers to O the government s collection of taxes and fees O the government s spending on infrastructures both the government s revenue and expenditures O All of the choices

What are the values of y and z A y O 8 D B N 38 C

A An nxn matrix B is a Square root of a matrix A if B A Show that the 2x2 Identity matrix 1 69 has an infinite number of real Square roots Adapt your froot from A to show that If ng 3 then then the nxn identity matrix In also has an infinite number of real Square roots

nd an orthogonal or unitary diagonalizing matrix for each of the following 1 a 3 i 3 1 4 b 1 1 1 1 1 1

You have two parents four grandparents eight great grandparents and so forth a If all your ancestors were distinct what would be the total number of your ancestors for the past 39 generations counting your parents generation as number one Hint Use the formula for the sum of a geometric sequence b Assuming that each generation represents 25 years how long in years is 39 generations yT e The total number of people who have ever lived is approximately 100 billion which equals 1011 people Compare this fact with the answer to part a What can you deduce Select all that apply some of my ancestors must not have had children There could not have been 39 generations in my line of ancestry People have been living on earth for less than 1 100 years Some ancestors on different branches of the family tree must be the same

Solve the following set of systems by the inverse matrix method x 4y 19 y 6z 8 2 2x 3y 5z 8 Step 1 of 3 Find the solution set for the first system of equations Answer x 4y 5 y 6z 29 2x 3y 5z 20 3 x 4y 9 y 6z 28 2x 3y 5z 17

e following matrix equation to find the matrix A 0 1 11 24 3 1 2 41 112 41

The vectors u and y have the same direction a Find u b Find v c Is u v Explain a lu b v c Is u v Explain Choose the correct answer below Simplify your answer Type an exact answer using radicals as needed Simplify your answer Type an exact answer using radicals as needed A No because the vectors have different magnitudes and the same direction OB Yes because the vectors have different magnitudes and the same direction C No because the vectors have the same magnitude and the same direction D Yes because the vectors have the same magnitude and the same direction 8 u TAV 4 4 4 4

Let S be the tetrahedron in R3 with vertices at X3 the vectors 0 0 02 and e3 and let S be the tetrahedron with vertices at vectors 0 V V and v3 See the figures to the right Complete parts a and b below a Describe a linear transformation that maps S onto S If T is a linear transformation that maps S onto S then the standard matrix for T written in terms of V V and V3 is A 1 b Find a formula for the volume of the tetrahedron S using the fact that volume of S 3 A 1 O C 1 2 det v det V V V3 det V V V3 area of base height OB det V V V3 V2 OD 1 det V V V3