Matrices & Determinants Questions and Answers

Find the state vector x3 for the given stochastic matrix A and initial state vector xo 9 16 13 31 4 15 13 31 1 3 3 19 16 19 XO

Use the matrices below to perform matrix multiplication A BA 5 6 2 2 B 1 1 1 42 5 1 2 7 2 2 1 2 49 4 8 3 6

Use the matrices below to perform matrix multiplication A 2 51 B BA B 6 7 8 8 6 8

Use the matrices below to perform the indicated operations 7 7 4 88 3 33 B C 6 5 A 5 2C B 7 2 5

Use the matrices below to perform the indicated operations 3 3 6 3 8 B 2 8 4 A A 6 8

question below Write an equation for the function graphed below 56 y 14 M 2 7 3 2 m 77 1 1 45 Question Help Video Message instructor

Perform the matrix addition or subtraction 4 12 12 B A A B 7 7

Use the matrix below to perform scalar multiplication 4 5 A 2 2 3A TIP

Use the matrices below to perform matrix multiplication 1 3 B A AB 3 1 3 4 2 4 71 5 3 2 7 5

Use the matrices below to perform matrix multiplication A 8 1 4 6 6 AB 5 6B B 5 7 3 7 5 8

If T P3 R M2x2 R is the linear mapping defined by Then a Rank T 0 b Rank T 1 c Rank T 2 c Rank T 3 a c dz b b c a b a c T a bx cx dx

Use the matrix below to perform scalar multiplication B 1 B 2 3 3 6 56 2 7 3 6 Preview

10 1 3 Let A 0 1 1 3 0 0 1 3 Numerically verify that each initial state vector xo has the given steady state vector x 0 58 Xo x Axo 0 21 0 05 0 74 0 35 0 28 0 37 0 42 0 0 535 A x0 0 465 0 Ax1 4 A x0 A x A x

5 pts Find the rank of this symmetric Nx N matrix It would be a tridiagonal matrix if entries positions N 1 and 1 N were zeroes A 2 1 0 0 0 1 0 2 1 0 1 2 0 0 0 0 0 1 0 0 0 2 1 0 1 1 0 0 0 1 2

4 12 pts Consider a matrix A 10 10 1 0 1 1 5 0 1 5 00 0 1 2 1 2 2 4 6 0 2 3 6 9 Suppose we have found its reduced row echelon form as 0002 4 a 8 pts Find its row column left null and null spaces as a span of their basis vectors b 4 pts Does A has a one sided inverse If yes find one

4 12 pts Consider a matrix A 0 1 0 1 0 1 0 1 1 2 2 4 6 2 3 6 9 Suppose we have found its reduced row echelon form as 0002 4 1 5 0 1 5 00 0 1 2 a 8 pts Find its row column left null and null spaces as a span of their basis vectors b 4 pts Does A has a one sided inverse If yes find one

281 1301 586 2711 PCP where P is invertible and C is of the form he matrix A has non real eigenvalues Find P and C so that

a b c d e 10 Which of the following is T 2 2 55 1 3 5 3 8 2 5 1 3 3 8 255 1 3 5 1 2 0 0 7 3 1 1 1 2 1 2 0 1 1 3 2 1 1 3 1 5 5 5 2 3 3 5 5 5

an orthogonal basis for the column space of matrix A is V V V3 Use this orthogonal basis to find a QR factorization of matrix A A 1 25 1 2 1 0 24 1 43 1 49 1 1 0 1 V 1 1 2 1 1 V3

For the following exercise find the domain and range of the function below using interval nota Domain 10 9 8 Range 7 6 9 5 A 3 32 2 24 10 9 8 7 6 5 4 3 2 1 1 23 3 56 5 6 8 9 10 1 2 3 6 7 8 9 IC Preview Preview

The graph of the relation z y 1 is shown Use the graph to find the domain and range and indicate whether its a graph of a function 6 5 4 3 2 Domain 00 00 Range 1 00 Is it a function yes 1 6 5 4 3 2 1 2 4 6 74 42 Preview Preview

3 A 16 1 20 60 22 41 18 51 66 121 120 60 120 901 a This matrix is diagonalizable The P matrix would be 17 12 6 17 12 61 47 2i 36 18 105 451 94 1 This matrix has eigenvalues 1 21 4 1

40 1 232 104 eigenspaces describe their basis Let A Write down the characteristic polynomial of A Find all eigenvalues of A and the corresponding

1 Determine the value of the unknown in the given matrix equalities 1 ii 5 2 13 x 2 36 4 X G 5 33 3 2 4 24 2 70 4 1 2

Prove that if A and B are similar then IA B Is the converse true Illustrate the results using the matrices 200 0 1 2 030 HECH P 1 12 p 1 1 4 2 2 2 5 A P AP 11 where B P 1AP STEP 1 First note that A and B are similar B 11 200 BBB 030 001 333 1 4 2 1 1 0 0 2 1 1 1 0 2 3 5 10 4 2 STEP 3 Does B A 8 6 4 4 10 7 000 000 STEP 2 Take the determinant of B by expanding along the first row B 3 5 16 10 012 1 1 2 2 25 8 101 4 7 10 1 1 0 0 2 1 8

Let F zy 5z 2y Use Stokes Theorem to evaluate J F F dr where I C is the curve of intersection of the parabolic cyliner z y z and the circular cylinde x y 25 oriented counterclockwise as viewed from above

Is Statment True or False Let T R R be any linear transformation Then 7 T TOT 10

termine whether the system corresponding to the given augmented matrix is dependent or inconsistent If it is dependent give the solute 10 9 3 201 8 6 00 0 0

he determinant of the matrix A 8 0 3 5 5 3 0 0 8 0 0

Determine whether the system corresponding to the given augmented matrix is dependent or inconsistent If it is dependent give the solution 100 4 010 4 000 5

Use elementary row or column operations to find the determinan 18 12 1 29 1

Solve the matrix equation A X B for X Let 860 347 A 1 2 5 29 6 and B Typo an integer or simplified fraction for each

A statistics professor is interested in finding out if there s a relationship between the final exam scores in Statistics 102 and the final exam scores of the same students who took Statistics 101 The data are shown in the table Stat101 Stat 102 87 91 68 72 46 95 82 73 75 84 95 63 73 60 90 90 62 74 Use a graphing calculator to find a line of best fit for these data Round to four decimal places Rounding to four decimal places the line of best fit for the data is y

Given the graph of y f x illustrated below sketch the graph of g x f 3 x 1 2 y f x NT

x g t dt where g is the function whose graph is shown below 2 Let G x g x A a Evaluate G 3 G 2 G 0 G 2 G 3 and G 4 b On what interval s G is decreasing c On what interval s G is concave downward d What is the absolute maximum value of G on the interval 2 4

The reduced row Echelon form of the matrix 1 2 1 2 3 0 0 1 2 is Select one 0 0 1 0 0 0 0 1 3 2 Pales 0 0 1 0 0 0 3 1 0 2 0 0 1 0 0 3 2 1 O None of these 0 0 1 0 01

3 2 1 2 D by elementary row operations Question 2 20 points a 13 Points Find the inverse of the matrix 2 3 A 1 2 2 1 1 1 1 0

3 7 points Let A 1 1 1 4 Find a singular value decomposition of A

Row reduce the matrix to reduced echelon form Identify the pivot positions in the final matrix and in the original matrix and list the pivot columns 124 1 245 5 454 10 Row reduce the matrix to reduced echelon form and identify the pivot positions in the final matrix The pivot positions are indicated by bold values Choose the correct answer below OA 10 0100 B 100 3 010 1 001 2 O C 100 1 010 2 001 1 D 100 3 010 001 1 0

Problem 3 Consider the Klee Minty problem covered in lecture maximize 10 1 j 1 subject to 2 1 10 100 1 for i 1 2 n x 20 for j 1 2 n a Write out the Klee Minty problem for n 3 b Using the matrix form of the simplex algorithm and the Python methods that we discuss in class solve a using the largest coefficient rule Make sure to print the dictionaries each step Submit both your code and its output

ulate det 45 A 1 2 1 0 17 1 1 20

The matrix Select one O True O False 1 7 5 5 0 13 2 1 2 3 if A 2 1 4 then A 1 3 4 1 Select one O True O False is in row echelon form 3 4 1 2 1 4 1 2 2 5 1 10 1 4 1 10 3 20

sing Cramer s rule find the equilibrium prices P P2 and quantities Q1 Q for two oods Qd Qal Qd 410 5P 2P2 Qa1 60 3P Qd2 Qs2 Qaz 295 P 3P Q2 120 2P

Given this frequency chart of 2201 passengers from the Titanic choose the class es whose relative frequency would comprise approximately 1 4 of a pie chart Class First Class Count Relative Frequency 325 1477 Second Class 285 1295 Third Class 706 3208 Crew 885 4021 Select one or more a First b Second c Third d Crew e None

Let A aij be a 3x3 matrix We know that 15 3 2 1 2 V2 2 1 1 9 0 2 are 3 vectors such that Av 1 2v 1 Av 2 6v 2 Av 3 9v 3 Find A and determine the value of a21 a22 a23

For the matrix A below find a nonzero vector in Nul A and a nonzero vector in Col A A 12 4 1 3 2 6 2 6 1 A nonzero vector in Nul A is www A nonzero vector in Col A is

For each matrix below you are given one of the eigenvalues Find a corresponding eigenvector for each eigenvalue 1 A 2 B 3 C 15 10 13 2 3 2 10 14 16 10 4i 6 2i 8 i 8 i 4 i 6 2i 10 25i 2 34i 12 124i 160360i and 1 A corresponding eigenvector would be 3 i 3 i 1 2i 16 22i 12 29i 52 108i 264 272i and 2 3i A corresponding eigenvector would be 26 44i 10 60i 74 205i 424 552i 4 20i 4 24i 1 47082 x 10 2 88i 86 267i and 4 i A corresponding eigenvector would be

Let A 1 4 9 2 2 3 a 15 points Find det A and det 2A b 10 points Find det A by reducing A into a triangular mat

Question 1 25 pts The augmented matrix of a linear system is given as 2 2 3 4 4 6 2 0 6 1 1 10 2 14 Reduce the augmented matrix into REF to find the possible solution or solutions for the system Show your detail work

A Matrix A and its characteristic polynomial are given Find if possible an invertible Matrix P and a diagonal Matrix D such that A PDP 1 Otherwise explain why A is not diagonalizable 11 1 on 00 07 0 1 0 5 ont 0 t 1 1 4 0 54 5 00