Matrices & Determinants Questions and Answers

2 What is the determinant of A 4 1 2 1 B 5 2 5

3 28 25 6 47 47 41 6 2A 4A 3I Let A 46 46 40 A 1 7 8 B 1 7 6 OC 1 7 10 D None of them E 1 7 12 which one is the eigenvector of matrix

points Let x y z be non zero vectors and suppose that z 4x 2y and w 8x 4y 3z Are the following statements true or fals 1 Span w y Span w x z 2 Span w z Span x y 3 Span w x Span w x y 4 Span w x Span w z 5 Span xz Span w y z

solve the systems of linear equations Enter your answers as a comma separated list If there is no solution enter NO SOLUTION If the system has an infinite number of solutions set wer and solve for x y and 2 in term 2x 3x 3x 4y 2w x 2y 2 Ex y z w 5

2 In Figure shown below assume no losses between section 1 and section 2 1 find a The discharge coefficient and the discharge under the gate b The force needed to rise the gate if the radius of the roller is 0 3m radius of the axis is 0 1m the coefficient of friction is 0 3 and k 1 5 T E L As the 1 2 Am H N ELIOM hva 0 6m EL 10m

tan 5x 12 cot 2x 3 find x value

1 Determine Jordan Canonical Form for the following matrice 1 00 00 a b INLO 3 4 2 1 12 00 0 0 1 2 0 12 0 07 000 100 1 2

5 Prove that if A is a symmetric matrix with eigenval ues A A2 n then the singular values of A are 121 122 n

points 2 Reduce the following initial value problem to one involving a first order linear differential equation system 4 points 1 y 3y 2y x e y 0 1 y 0 y 0 3 2 t t b Please find the corresponding fundamental matrix and solve the above linear differential equation 4 points c Please find the corresponding et 4 points d Please prove the linear differential equation system solution derived using et will be the same as b 4 points e Please also use Laplace Transformation method to solve the problem 4 points

6 16 points Suppose A 0 1 1 0 2 1 b 2 0 2 1 Find a full rank decomposition of matrix A

For what value of K do the simultaneous equations 2x 3y 1 4x 6y K have infinite solution Ok 3 Ok 4 Ok 2 Ok 1

Given X1 JER X 20 X E R x 0 X3 Is S a subspace of R3 Explain your answe S

Let L R R be defined by i 20 P ii 05 P X 2x y 00 y x 2z Determine whether L is a linear transformation Find the standard matrix representing L

A basis of R is given as B x 1 1 x 1 0 Find the change of basis matrix P from basis B to ba B where B y 4 3 y2 3 2 o p OP OP o P 4 3 4 1 8 2 5 2 1 8 1 2 3 2

15 1 point Solve each equation or inequality Check your sol 1 0 O O O O Q5 3 2x x 7 0 x 7 x 0 Previous

Use implicit differentiation to find y 5x 2y 4x dy dy dz

3 10 marks Let V P 1 1 be a inner product space with inner product defined by f t g t dt for f 9 V Find the polynomial of degree one or less in V that is closest to h x sin x in V

5 a 8 points Find the best least squares line C Dt to fit the data points 0 1 1 3 2 2 3 4 and 4 5 b 2 points Sketch a graph of the data points and your least squares line c 2 points Find the least squares error e of the best fit line

Question 11 Let f be the reflection transformation in R2 about the line d 3x 2y 0 Find the transformation of f with respect to the standard basis A w

Let P R4 R4 be the orthogonal projection onto the hyperplane W x y z t 2x y t 0 That is the projection parallel to the normal vector 2 1 0 1 Find the matrix ME P of the projection Prelative to the standard basis of R4 ME P 0000 000

and 1 and 8 Find the eigenvalues and eigenvectors of the matrix A 11 3 2 2 12 0 3 0 3 0 11

Let A 41 0 Find a matrix B such that 3A 2B is a scalar matrix with det 4B 6A 8 1 2 12 5 1 0 4 3

With the aid of graphs explain whether or not the inverses in question 2 are functions

20 points Find the QR factorization of the given matrix 4 6 12 20 9 3

4 20 pts Suppos the motion of a particle on a plane can be described by the equtions x t 5t 3t 11 y 1 2t 3 Find the point where the particle crosses its previous path find the x y coordinates of this point and the t moments in time when this happens What is the velocity of the particle at these times What is the distance the particle travels to complete this loop You may want to sketch the curve to help your exploration

20 points Find the QR factorization of the given matrix 6 2 3 3 20 9 00

X1 3x3 3 2x ax x3 2 igib dilin x 2x ax3 1 PL 25 y god linear equation system is given Determine value a a No solution b One solution in C insbura hoy aniel E 2 c Infinitely many solutions 25p

Consider the following system dx dt dy dt Find the eigenvalues of the coefficient matrix A t Enter your answers as a comma separated list 6x 5y K 2x 8y Find an eigenvector corresponding to the eigenvalue with positive imaginary part Find the general solution of the given system x t y t

For the pair of similar triangles find the value of x 26 8 The value of x is Simplify your answer Use a comma to separate answers as needed

00 00 06 Question 1 of 60 of the 13 members of the Texas Real Estate Broker Lawyer Committee how many must be appointed by the Texas Real Estate Commiss OA All O B 6 OC S OD 10 4

using the standard Form of a linear Equation Ax By C What is the slope

5 1 14 001 4 Determine the dimensions of Nul A and Col A where A

log x b c appears below a 0 2 1 2 5 x b 1 Submit Question Show Transcribed Text 4 3 3 Write an equation for the transformed logarithm shown below 12 1 5 4 3 2 1 2 3 C 4 3

1 5 points Determine the matrix e for J 0 00 and A arbitrary

3 calculate the volume of solid object which is limited by a b c d according to the picture given a y x y z 4 x 0 z 0 b 1 z y y x x 0 c x y z y y 1 x 0 dy v2 2 V A 3v 47

se spherical coordinates to integrate f x y z 4 x y z over the solid E above the cone z 3x 3y and side the sphere x y z 4 Enter an exact answer Use symbolic notation and fractions where needed 4 z dV

The equation 5x2 6x3 0 is given Firstly Then find the solution set Find a basis for the solution set Explain why the solution set is a subspace of R3

11 Let A be a commutative ring and M an A module Define the support of M by supp M p e spec A M 0 If M is finite over A show that supp M ann M where ann M is the annihilator of M in A that is the set of elements a E A such that aM 0

please show all the steps and explain I am more concerned about the explanation metrology than the answer please write clearly Find all solutions to 21 223 4 8 12 3x3 21 6 321 422 623 824 0 ro 3rr 4r 12 Seme

1 70 to buy the supplies to make a fork and 1 30 to buy the supplies to make a spoon 2 Molly s Custom Kitchen Supplies sells handmade forks and spoons It costs the store The store sells forks for 5 60 and spoons for 5 40 Last April Molly s Custom Kitchen Supplies spent 37 90 on materials for forks and spoons They sold the finished products for a total of 147 20 How many forks and how many spoons did they make last April a Define your your variables and set up the system of equations in in Standard From b Rewrite your system of equations in Standard From using your original variables then rewrite the system replacing your variables with X and Y Write your matrix and solve with calculator Write system with new variables here let F represent let S represent let X replace let Y replace Set up matrix and solve with calculator here

Part 1 of 2 Factor f x given that 1 is a zero f x Part 1 2 Part 2 of 2 X The solution set is Solve f x 0 Express your answers in exact simplest form 8 0 0

If T is defined by T x Ax find a vector x whose image under T is b and determine whether x is unique Let A 2 Find a single vector x whose image under T is b 1 6 19 4 GEEN 6 and b

point Given 2 3 2 8 2 find the closest point to in the subspace W spanned by 4 and 6 2 2 24

2 Reflect the image over the y axis and then translate 4 the coordinates of the mapped image E E 2 5 3 2 1 F H 2 3 s F 12 1 2 3 4 F G G H C4 D 142 H

rthogonally diagonalize the matrix giving an orthogonal matrix P and a diagonal matrix D To save time the eigenvalues are 5 1 and 1 2 2 1 A 2 1 2 1 2 2 nter the matrices P and D below

4 20 points Let L be the line given by the span of 7 in R Find a basis for the orthogonal complement Lof 4 A basis for Lis

Let W be the set of all polynomials of the form ast a3t a t a t ao where a2 0 and ao 2 Then Select one W is a subspace of P4 and a subspace of P3 W is a subspace of P3 but not a subspace of P4 None of them W is not a subspace of P4 and not a subspace of P3 O W is a subspace of P4 but not a subspace of P3

Let TA R R be multiplication by the orthogonal matrix 1 2 3 3 3 2 A 3 2 3 2 3 2131130 31722 Col 3 3 12 Find T x for the vector x TA x 222 01 3 7 Using the Euclidean inner product on R verify that T x T x x

Problem 2 Find an orthogonal matrix P that diagonalizes 4 2 2 A 2 4 2 2 41 12 Then write the spectral decomposition or eigen decomposition of A Problem 3 Optional What is a geometric interpretation of a spectral decomposition

For the matrix A find an orthogonal matrix P such that P APis diagonal 3 6 0 6 0 0 6 1 13 2 6 3 Show Transcribed Text 2 1 2 11 2 1 2 C 2 2 1 2 2 1 2 C 7 2 2 2 1 1 2 2 2 1 2 2 1 2