Matrices & Determinants Questions and Answers

Consider the following 4 05 6 16 5 04 a Compute the characteristic polynomial of A det A 2 2 A b Compute the eigenvalues and bases of the corresponding eigenspaces of A Repeated eigenvalues should be entered repeatedly with the same eigenspaces 2 N 23 li has eigenspace span has eigenspace span has eigenspace span 1 11 11 c Compute the algebraic and geometric multiplicity of each eigenvalue A has algebraic multiplicity and geometric multiplicity 22 has algebraic multiplicity and geometric multiplicity 23 has algebraic multiplicity and geometric multiplicity smallest value largest value

Compute the indicated 3 x 3 determinant abo 0 a b

Determine if the matrix below is invertible Use as few calculations as possible Justify your answer 1 4 7 4 3 5 0 0 2 SCOOB Determine if the matrix below is invertible Use as few calculations as possible Justify your answer OA The matrix is not invertible In the given matrix the columns do not form a linearly independent set OB The matrix is not invertible If the given matrix is A the equation Ax b has no solution for at least one b in R OC The matrix is invertible The given matrix is not row equivalent to the nxn identity matrix OD The matrix is invertible The given matrix has 3 pivot positions

4 What elementary row operations applied to 3 1 1 27 2 30 1 12 will yield the following matrix 3 5 2 30 0 3 1 2

7 10 points Apply Cramer s rule to solve the following linear system 21 1 1 2 2x x 50 7 2r 0 Use any formula of choice to compute all the associated determinants Show all your work

Let A u v w be a matrix with columns u 12 0 3 0 W

Two augmented matrices for two linear systems in the variables x y and z are given below The augmented matrices are in reduced row echelon form For each system choose the best description of its solution If applicable give the solution b 10 2 01 0 LO O O 10 1 012 000 8 0 The system has no solution O The system has a unique solution x y 2 0 O The system has infinitely many solutions x y z x x x D x y z 1 The system has no solution The system has a unique solution x D The system has infinitely many solutions x z x D C x 2 x x x 2 00 X

3 Calculate the convergence Incos 1 n not b sin Than t

the matrix elect one 210 a 2 R R Ob 2 R R OC 2 R Od 2 R R 2 R R was obtained from by an elementary row operation Which of the following describes the elementary row operat

16 Let A X 9 1 1 y 4 270 If the kernel of A contains the vector what are x y and z 2 3

of Which of the following is the reduced row echelon form for the augmented matrix of x 2yz 8 x 3y 2z 3 2xy 32 19 Select one a g this question for future reference 001 O b OC O d e 00 01 0 3 0 0 100 010 001 0 20 1 0 0 100 010 0 00 3

nonzero matrix has an inverse tone ue Ise

The augmented matrix in echelon form of a system of linear equations in x and y is given below Find x and y 33 Enter the exact answers X i

In the following exercises express the limits as integrals n 60 lim x Ax over 1 3 i 1 n 61 lim 5 x 3 x A over 0 2 3 n x i 1 n 62 limsin 2x Ax over 0 1 i 1 81x

Find the inverse of the function on the given domain X f x f x 2 812 18 00 RHD

Exercise 3 6 7 Let a e3 a2 e and a3 e2 be the vectors of the ordered basis A for R a Find the transition matrix S for the change from the standard basis to A b use S to find the components of x 3 4 5 in the basis A and c find the representation MA of the matrix M 1 1 1 2 2 0 3 0 0 relative to the basis A

Question 6 entify the x intercept of the function y 4 The function has no x intercept 1 81 4 81

Question 1 Evaluate the function f x 3 44 may use your calculator

formulas for the entries of M where n is a positive integer M 2 M 3 4 8

automatically If logg 81 x what is the value of x

sider the following B 1 2 1 1 B 12 0 4 4 1 x g a Find the transition matrix from B to B b Find the transition matrix from 8 to B P c Verify that the two transition matrices are inverses of each other Pp 1 LI d Find the coordinate matrix x g given the coordinate matrix x

Find all steady state vectors for the given stochastic matrix A X 0 09 0 08 0 09 part b 0 08 0 Show Transcribed Text Fill in the missing values to make the following matrix a stochastic matrix 0 35 0 54 0 06 0 27

2 A 50 meter vertical tower is braced with a cable secured at the top of the tower and tied 30 meters from the base What angle does the cable form with the vertical tower

Question 24 3 3 points Using a Standard Normal Distribution what z score has an area of 892 0 108 0 814 A

Find the points on the graph of the polar equation where the tangent line is horizontal or vertical Assume 0 0 2 Enter your answers as a comma separated list of ordered pail r 1 cos 0 horizontal tangent smallest 8 value argest 8 value wertical tangent smallest 8 value argest 8 value r 0 r 0 r 0 r 0 r 0 r 0 117

6 Find the least squares solution to ther overdetermined system Ax b Also find the orthogonal projection of b into Col A A 1 1 1 1 3 1 b 3 0 8

find the standard matrix for the reflection T of R 3 in the line x 2t y t z 2t Is T invertible

Use the factorization A PDP 1 to compute A where k represents an arbitrary integer 19 a 0 1 9 01 0 b 0 1 a 9 b a O

Diagonalize the following matrix 400 140 005 Type an integer or simplified fraction for each matrix element OC The matrix cannot be diagonalized

1 point Find a basis of the subspace of R4 that consists of all vectors perpendicular to both Basis 1 0 7 5 and 0 1 6 5

Find a power series for the function centered at c 1 9 x f x 8 n 0 f x I C 2 Determine the interval of convergence Enter your answer using interval notation

11 Only one of the following matrices has an inverse Find the determinant of each matrix and use the determinants to identify the one that has an inverse Then find the inverse 1 4 1 0 1 0 1 A 0 2 B 14 0 0 20 301

Let S 1 2 3 and f1 f2 f3 Fun S be given by Find a nontrivial linear relation among these vectors f f f3 0 fi 1 2 fi 2 0 f2 1 2 f2 2 6 f3 1 3 f3 2 9 f3 3 9 f 3 5 f2 3 4

60 Question 4 14 2 points In the following data set what word would best describe the data entry 10 10 18 18 19 19 20 21 21 23 23 24 27 Bias O Outlier O Range O Mean Question 5 14 3 points Consider the following histogram in which a student created a histogram about the heights of males in his classroom 5

tra Credit Add or Subtract Write your answer in standard form 1 4 coefficient and degree 1 2x 3x 9x 2x 8x 11 s completely 2 16p 4 4 4x 8x 4x 20 HW 12 Subtract from

The below permutation matrix P changes the second and third rows of matrix A when A is multiplied with P PA Find the unknown values in P P f 1 d e h b 1 a C 2 i O a b c def g h i 1 0 2 4 g 0 47 1 46 44 4 O

Let T R R be a linear transformation such that T 1 1 1 3 0 1 T 0 1 2 3 4 1 and T 1 0 1 0 1 1 Find the indicated image T 2 1 1 T 2 1 1

Find a singular value decomposition for each of the following matrices 9 a 2 b 0 1 0 0 02

After a person takes a pill the drug contained in the pill is assimilated into the bloodstream The rate of assimilation minutes after taking the pill is R t te 0 4t Find the total amount of the drug that is assimilated into the bloodstream during the first 15 minutes after the pill is taken Round your answer to 2 decimal places 0 23 6 25 0 38 6 14

A 6 1 5 3 2 8 2 1 7 4 6 7 7 6 55 10 6 3 6 Enter the singular values of A in decreasing order into the answer box below separated with com

4 Solve each triangle Round your answers to the nearest tenth of a unit a b T 5 cm 7 cm R 9m S U 10 m c In AHIJ LI 90 i 1 5 m and ZJ 21 P d In ABCD LC 90 b 12 0 cm and d 13 5 cm e In ALMN LM 90 1 9 6 m and LN 34 2

A P 1 1 where Pis the initial amount The equation for compound interest is A P invested r is the interest rate as a decimal n is the number of times compounded annually and t is the number of years A person is investing 1000 at an interest rate of 12 interest for 25 years and is curious how much difference the number of compounds n increases the value of the account after 25 years She created the function A n 1000 100 1 0 12 25 What would best describe an appropriate domain of the function The set of Integers i e 2 1 0 1 2 The set of Natural Numbers i e 1 2 3 4 The set of All Real Numbers R where 20 A

Consider the function f x W Oy 0 None 4x find any horizontal asymptotes if they exist 3x 2

3 12 12 3 6 6 6 with rational entries and a diagonal matrix D such that D PT AP Let A P 100 100 6 6 Find an orthogonal matrix F D

Determine the type and stability of each real equilibrium point by calculating the Jacobian matrix at each equilibrium a b c x 1 xy y x y x x 3y 2xy y 4x 6y xy x 4x x xy y 4y x y y

Let 1 A 3 5 Find bases for Row A Col A and Null A 2 1 5 3 6 1 15 2 10 1 25 4

Graph the functions 10 y 3 2x 6 5 4 3 2 1 0 1 2 3 4 5 6 x TIG 7 1 a at w No to L T 7 11 f x 4 x 1 MEE 6 5 4 B 1 I YA 2x O 1 2 3 4 5 6 x 23 CHIL CHIL nit F 12 g x 2 E 6 5 4 3 LLL J 1 56 ol 1 2 3 4 10 5 6 x

4 6 y 2 and u 3 3 Compute the distance from y to the line through u and the ori

Scalene Triangle A MATH MONKS B C If the Measure of ZA 94 and the measure of ZB 91 what is the measure of ZC

If A A 1 1 0 0 a 1 b 0 0 find det Adj A 1