Matrices & Determinants Questions and Answers

M3 2 53 M3 1 14 C2 1 17 M1 3 21 M1 2 7 M2 3 18 C2 2 6 C3 3 38 a3 2 3 a3 1 0 a2 1 7 a1 3 5 a1 2 2 a2 3 3 a2 2 4 a3 3 1 Calcule el determinante de A

Let v 3 and vg 3 Let v 4v1 2vg Find Av 1 be eigenvectors of the matrix A which correspond to the eigenvalues A 1 and A 4 respectively

Question 2 Let V R Show that W is not a subspace of V where i W a b c a 0 ii W a b c a b c 1

At Suburban Community College 40 of all business majors switched to another major the next semester while the remaining 60 continued as business majors Of all non business majors 30 switched to a bu major the following semester while the rest did not Set up these data as a Markov transition matrix HINT See Example 1 Let 1 business majors and 2 non business majors Calculate the probability that a business major will no longer be a business major in two semesters time

ine sales of certain products in three cities are given in the 4 by 3 matrix below City 1 City 2 City 3 68 O 57 75 74 33 77 53 73 The cost of the items depend if they were sold Retail or Wholesale Tablets Laptop Computers Cell Phones Personal DVR 52 46 60 59 The Retail costs were 186 for tablets 307 for laptops 362 for cell phones and 1 121 for DVRs The Wholesale costs were 294 for tablets 473 for laptops 235 for cell phones and 1 121 for DVRs To get the total revenue at each city for Retail and Wholesale we would need to put the information above on retail and wholesale costs into a matrix that has rows and columns and multiply it by the 4 by 3 matrix on sales at each city for each item After multiplying these two matrices we would have a matrix that has columns The total wholesale revenue from city 2 would be in row and column rows and

Find the limit if it exists of Ax as n approaches infinity for the following matrices If an answer does not exist enter DNE 01 A and x Ilm Ax 1 00

Consider the following sets ER 7x 6y 5z 0 6y 5x 0 B 0 C pe P3 P 2 p 5 5 0 A R3 Select all of the following which are true A is a subspace of R3 B is a subspace of R OC is a subspace of P3 3 and 22 4y z 2

1 point Find a basis of the subspace of R consisiting of all vectors of the form Basis X1 5x1 x2 8x 6x 9x1 7x2 You should be able to explain and justify your answer Enter a column vector such as 2 using the syntax 1 2 3 If there is more than vector in your basis enter a comma separated list such as 1 2 3 4 5

Let T R 4 R 3 be a linear transformation Find T 1 3 and T if T 3 5 14 0 2

In Exercises 15 18 find the spectral decomposition of the matrix 3 1 6 5 1 2 3 2 15 16 2 3

Find the least squares solution of the linear equation Ax b and compute the least squares error where A 3 2 1 1 4 3 1 10 7 9 2 and b 2 14 points

a11 a12 a21 a22 Which of the following is are TRUE Let W Select one or more a W is not closed under matrix scalar multiplication b 8 0 W 0 C w 0 W M2x22 R aij aji fori j d W is an empty set e W is not closed under matrix addition f W is a subspace of Man R

Use the graph to write the formula for a polynomial function of least degree f x 6 4 2 f x 6 I 4 2 2 4 6 2 4 6 X

marks 3 d Consider the following network of irrigation channels with flows mea sured in say thousands of litres per day B 9 A X1 X3 2 X2 X4 9 Set up a linear system for the possible flows x1 x2 x3 x4 Write down the corresponding augmented matrix for this system but do not solve it

1 1 2 Given that the determinant of 4 22 235 3 is equal to 33 find the determinant of the matrix 1 4 0 1 2 0 2 0 0 10 1 5 3 0 0 2

5 Let A1 9 42 2 2 x 2 matrices Suppose that 4 1 3 13 43 3 3 1 01 E 0 E2 0 E3 10 E 09 For each of the following select True if the statement is correct and select False if otherwise a The dimension of span A1 A2 is 2 True False 3 8 42 be matrices in M22 the vector space of all b The dimension of span A1 A2 A3 is 3 True False c The set A1 A2 A3 is a basis for M22 True False d The set A1 A2 A3 is a basis for span A1 A2 A3 True False e The set A1 A2 is a basis for span A1 A2 A3 True False f The set A1 A2 E2 E3 E4 is a basis for M22

matrix Let a 1 and b 0 A b a a b a b b a

The graph below shows at least one complete cycle of the graph of a trigonometric function Find an equation to match the graph 6 5 3 2 7 3 4 5 6 2 75 2 5 2 25 N 1 75 1 5 1 25

The row strategy is R 1 2 1 2 and the Column strategy is C 1 4 and the Payoff matrix is P 1 3 3 4 45 Find the expected value for this information 9 8 9 8 3 8 3 8

Find a basis for the subspace of R spanned by S S 5 9 9 1 2 2 1 1 1 STEP 1 Find the reduced row echelon form of the matrix whose rows are the vectors in S 0 0 0 0 1 0 11 1 0 STEP 2 Determine a basis that spans S 0 0 1 0 11 31

Diagonalize the matrix by finding a nonsingular matrix P and a diagonal matrix D such that B PDP 1 A 13689 26877867 B B PDP 1 6969 PDP 3 9 0 31 0 1 2 1 2 1 2 1 2 PDP 16197 2

2 2 0 2 Decide which of the following if any are an eigenvector of the matrix A above If the vector is an eigenvector of A find the associated eigenvalue a 2 4 6 b 2 0 6 c 2 2 0

Use the inner product f g f z g z dz in the vector space C 0 1 to find 1 9 and the angle of between of and g for f x 3 and g x 9x f g f g 019 radians

et W be the subspace of R spanned by the vectors 18 8 81 Eles 10 10 2 3 colt 3 21 0 Find the projection matrix P that projects vectors in R onto

The columns of Q were obtained by applying the Gram Schmidt process to the columns of A Find an upper triangular matrix R such that A QR A 2 3 7 5 2 2 4 6 2 7 57 5 4 2 7 2 4 2 7

Find a 2x2 matrix A such that Av w Av W where v 2 5 w 3 2 W v 3 2 5 2

1 Which of the following matrices are in reduced row echelon form Circle all that apply 103 10400 01300 e 0 0 0 10 00001 00000 009 a 0 0 8 017 0 1000 b 0 1 0 0 00 10 10003 c 00102 0001 1 d 10900 01000 00100 00010

X Find the value of 3u X y 3v 3u 2 N 3w 7 N 11 y 3v 3w if it is known that 1 2 7 N x y 1 2 7 5 U V W

et nd AR 1 1 3 G 9 2 4 11 3 A B 3 2 2 4 0 7 0 7 5 0 2

Find all values of the a for which S 2 0 3 2 2a 4 0 3 0 a is a basis for R A B R 3 2 2 5 R 2 2 5 1

Find Q s t Q AQ J where 1 J is the Jordan form A A simplify of 100 A

Find a particular solution of the indicated linear system that satisfies the initial conditions x 0 3 x 0 5 and x3 0 7 4 H 3 21 24 0 X 18 21 0 X X 18 18 3 3t 3 X 3t 1 1 X31 1 The particular solution is x t x t and x3 t 3t 1 0 I thought I knew what to do and got a totally wrong answer

Q1 7 marks Let V span a a2 a3 CR where 0 0 0 4 5 a Use the maple output at the end of this test paper or otherwise to prove that the vector b b b b E belongs to the vector subspace V if and only b3 b 2b2

a e b Let A 10 e b Under what condition will these eigenvalues be real numbers

5 15 Marks For what value of k is the system 1 3 2 1 2 1 2 3 5 4 4 3 10 2 1 2 X1 X2 x3 X4 13 5 7 k consistent Determine the general solution with this value k

Consider the transition matrix is equal to If this is a transition matrix for a Markov chain with initial state vector xo O 200 150 350 320 180 200 280 300 120 WIN W10 700 0 P 0 lim P xo 100 200 then the limit 400

Consider the transition matrix is equal to O 200 150 350 320 180 200 If this is a transition matrix for a Markov chain with initial state vector x 280 300 120 P 700 0 11 1330 2 3 les les O lim P xo 11 0 100 200 then the limit 400

x y z 2 2x y 5z 1 x 6y 4 over Z7

Let T R R be the linear transformation that first rotates points clockwise through 135 3 4 radians and then reflects points through the lin y x Find the standard matrix A for T A

Determine if the statement is true or false and justify your answer If det A 0 then the columns of A are linearly independent True by The Big Theorem False For example if A False For example if A False For example if A False For example if A 00 00 10 00 1 1 0 11 01 then det A 0 but the columns are linearly dependent then det A 0 but the columns are linearly dependent then det A 0 but the columns are linearly dependent then det A 0 but the columns are linearly dependent

Let A 0 1 2 2 3 4 1 4 3 12 1 2 Find PLU factorization

Let A 7 2 5 3 v 06 9 1 9 10 62 Find a nonzero vector in Col A

The matrix A eigenspace 0 15 0 14 0 0 0 14 The eigenvalue is 14 A basis for the eigenspace is has one real eigenvalue Find this eigenvalue and a basis of the

For the matrix A find an othogonal matrix P such that PAP is diagonal 3 2 4 A 2 6 2 4 23 a 2 3 3 2 1 3 1 3 1 2 2 3 2 2 3 2 1 3 2 4 3 2 1 3 2 1 3 2 4 3 2 1 1 2 3 2 b d 2 3 2 2 3 2 3 1 3 2 3 2 1 1 2 3 2 1 2 3 2 1 1 3 2 4 3 2 1 1 2 3 2

13 2 points Diagonalize the matrix 2 A 1 1 5 2 1 1 3 1 2 2

Given matrix A 2 1 302 10 Find the co factor matrix of matrix A Use your result from part i to find inverse A

Let us consider the matrix A 23 4 1 i Find the eigenvalues and their corresponding eigenvectors of AX XX ii What are the invertible matrix P and the diagonal matrix D such that A PDP iii Find the inverse P 1 iii Use your results from parts ii and iii to calculate A

Use the standard algorithm to compute the quotient a 545 107 b 1045 253

Question 3 Compute A 604 2 7 1 312 A

olve for X 8 73 x 1 X X 5 4 2 8