Matrices & Determinants Questions and Answers

Let A QR be the QR decomposition of A where A QR 1 2 0 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 0 0 1 2 0 1 2 Determine the minimum possible value of Ax b where 1 1 0 b 0 0 0 0

Which of the following is the trace below 00 01 10 trace AB7 where A CA 0 0 and 8 2 1 2 B 0 10

tx 5 H 2 3 and y yll yll yll 3 3 Compute

Let C 1 a b b Compute QQ and so find Q if Q 0 Q a b c ba d C d d C c d a

Let A 123 1 28 and v G Also let H be the two column vectors of A Compute A and A a Av b 44 5A What do you notice about these answers 28

Consider the linear system 22 12 10 2 a 142 a For which value of a will the system have no solutions b For which value of a will the system have infinitely many solutions case Use t as a parameter for your solution z t y t Write down the set of solutions in this D c In the other cases the system has a unique solution This solution depends on the value of the variable a and is equal to B

For k 0 use the row reduction method to find A 1 A if given 12 0 13 1 13 k 1

1 point Let T R R be a linear transformation such that 2 0 9 0 4 0 0 T Find the following 7 0 8 3 7 Hint First write 5 3 as a linear combination of 9 400 and 1

1 For k 0 use the row reduction method to find A if given 1 2 0 3 1 3 k 1 A 1 1

Let 1 3 0 1 k k 0 31 Find the values of k for which A is not invertible NOTE Enter all values exactly Number of answers 0 A

Consider B ir Ma 3 what is Me B 1 and 1 A ve vc B M 2 ve 11 22 0 v c 3 C are bases for R2

Find a basis for the nullspace of the matrix If there is no basis enter NONE in any single cell 13 6 4 8 6 3 2 4 2 1 1 2 4 2 2 3 A

Let 4 1 2 B 2 C 113 1 D 3 Evaluate each of the following if possible If it is not possible justify your answer Show all your work CET 3E 13 tr D B Find the first column of BA a 3 marks b 2 marks c 4 marks d 5 marks 1

Let u 3 2 3 and v 2 4 2 Find ux v

Consider A What is the maximal order of a minor A 1 B 3 C 4 5 0 1 5 2 3 4 1 2 1 3 3 1 D 2

Let A 272 1456 Decide whether the matrix A is invertibile and if so use the adjoint method to find its inverse Enter as a matrix exactly in fractional from if required if not invertible enter NA

These points are linear Find the slope x 3 2 1 012 y 3 2 10 12 slope Enter

Let A A 11 If the matrix is invertible use the adjoint method to find its inverse If the matrix is not invertible indicate that using the checkbox NOTE Write the elements of the matriz exactly 2 2 0 0 0 0 2 1 2 0 10 14 T 121120 3 20 10 0 X

1 point Let Define the linear transformation T R R by T A Find the images of u H 4 T T u 0 7 0 BH A 0 02 800 T v 4 and 7 a 1 C b under

5 In the diagram the inscribed angle LABC has a measure of 50 AB BC and the radius of the circle shown is 10 units What is the length of AB A 357 16 B 657 6 C 657 9 D 42n 5 B A C

If V V2 V3 is a basis of R3 show that U 42 433 is also a basis of R3 U V U V V U3 V V3 2 The column space and null space of a of 7x10 matrix A are subspace the same dimension Find nullity AT

Use Cramer s rule to solve the system y z 4 X y 2z 14 x 4y 2z 17 X Find the determinants D x 0 0 0 0 Select the correct choice below and fill in any answer boxes within your choice OA The solution set is Simplify your answers OB The system contains dependent equations OC The system is inconsistent www

Problem 13 7 points Find the eigenvectors for the matrix A B 1 1 9 6 D C C E 4 0 1 03 0 102

c Find a basis for the solution space for AX 0 where A What is the dimension for this solution space 1 2 2 2 1 1 1 1 2 0 1 3 1 3

Let A PDP 1 and P and D as shown below Compute A4 P A4 13 25 D 1 0 03

A p 2 B F 3 Given A use the Frobenius inner product and the corresponding induced norm to determine the value of each of the following A B and B 5 2 4

1 point For which value of k does the matrix A 58 5 6 have one real eigenvalue of multiplicity 2 k

below Thanks 1 Confirm by multiplication that is an eigenvector of A and find the corresponding eigenvalue 1 2 1 X 32 a A 4 0 1 b A 2 32 104 X 3 2 1

2 Solve the system of algebraic equations x 2x 0 X 2x x 1 3x x x 6 by Cramer s method Check the answer

a Solve the following equation for matrix X 0 0 0 P 000 1 1 0 23 RX B 3 x 2 12 0 1 1 23 12 1 1 1

3 Construct an orthonormal basis for the subspace of R4 W span x1 x2 x3 15 x 0 3 4 0 C x2 70 2 1 2 C X3 1 0 0 0

1 point Which of the following sets of vectors are linearly independent Check the boxes for linearly independent sets 8 0 A B 0 7 5 1 2 4 2 D 5 3 1 8 4 C 6 5 0 0 0 4 5 1 2 7 4 G E 2 7 8

Which of the following is NOT in the span of x and y 2 1 1 1 3 8 2 3 H 3 5 Y

Find the rank and nullity of the matrix then verify that the values obtained satisfy Formula 4 in the Dimension Theorem 4 69 24 10 11 0 2 4 2 4 5 13 21 30 A 2015 rank A nullity A rank A nullity 4

For the encoding matrix 13 2 5 3 10 A 29 54 6 19 decode the message GE 42 109 433 198 181 which gives a six character word 797

Question 2 4 points Find a QR factorization of A 1 1 1 2 2 2

2 1 13 52 26 find an orthogonal basis under the Euclidean inner product Let 3 112 Orthogonal basis V x a Ex 1 23 113 3 V 3 b Ex 1 23 33 22 26 V3 c Ex 1 23 be a basis for R Use the Gram Schmidt process t a B b

Let W Span x1 x2 where 1 and 1 3 1 3 2 3 Compute the Orthonormal Basis Using Gram Schmidt Process

Express each column vector of BB as a linear combination of the column vectors C C2 and c3 of B 3 4 2 B 0 1 6 5 Enter first column as a linear combination of columns of B in terms of the vectors C C2 and c3 Enter second column as a linear combination of columns of B in terms of the vectors C C2 and c3 Enter third column as a linear combination of columns of B in terms of the vectors C C2 and c3

Use Gaussian Elimination to carry the following matrix to a REF 2 31 2 4 5 1 1 4 2 3 7 What are the positions of the pivots 1 2 1 2 1 1 1 2 3 and 3 4 1 1 and 2 3 1 1 2 2 and 3 3 1 1 2 2 3 3 and 4 4 1 1 2 2 and 3 4

1 Let A a n be symmetric and positive definite Show that a a 0 i 1 2 n b a aa i j 1 2 1

1 A network of six one way streets A B C D E F is shown in the following diagram where 100 cars per hour enter at streets A and B 200 cars per hour enter at streets D and E 50 cars per hour exit at streets A and C and 250 cars per hour exit at streets E and F 100 A 200 B 50 C D E 250 a 4 marks Represent the flow of cars in this network using a system of linear equations and represent the system using an augmented matrix b 5 marks Identify the rank of the system and find a possible solution

What is the image of X 1 A 2 32 B 2 under the linear transformation T where I is the reflection across X2 X1

The linear transformations T R R and T R R are defined by T1 X EDH X3 T X2 What is T T 2 W A 2 B 7 X1 X3 X2 X3 X3 X1 3 X1 CH X3 X1 X1 X2 X1 x x3 and

8 Find the points on the surface z xy 1 that are closest to the origin using LaGrange multipliers

7 Solve the inequality set Graph the solution on a n 14 x 5 O x 9 or x 19 O x 9 or x 19 O x 9 or x 19 O x 9 or x 19 O all real numbers no solution

thy the ver ex or Remember y ax bx c O 9 48 O 4 31 O 7 84 O 3 4 by 10

Using the discriminant state the number of soluti For the following equation 3y 18 one real root two real roots three real roots O no real roots Using the discriminant state the number of soluti for the following equation 4x x 5 one real root two real roots three real roots

Solve the system if possible by using Cramer s rule If Cramer s rule does not apply solve the system by using another method Write all numbers as integers or simplified fractions 2x 6y 3z 2 4x 12y 6z 4 6x18y 9z 6 Part 0 2 Part 1 of 2 Evaluate the determinants D D D and D

Use the Gram Schmidt process to produce an orthogonal basis for the column space of matrix A An orthogonal basis for the column space of matrix A is Tyno a voctor or list of woctore Ilco o con congrato vectors of nooded A 9 1 7 16 1 9 15 16 1 9 23 3 5 6 21 15 13 6 3 O