Matrices & Determinants Questions and Answers

Perform the indicated operation 1 2 5 3 4 8 X 0 10 14 6

Find 2A if A 2 7 3 1 5

Use row reduction to solve the linear system X1 2x2 3x3 6 2x1 3x2 2x3 14 6x1 2x22x3 2

Kiran and Mai are trying to figure out if this equation is an identity a b 4 a 4a b 6a b 4ab b Kiran picks several different values for a and b to test For each pair he tries the left side and the right side have the same value Since his tests all worked Kiran thinks the equation is an identity Who do you think had a better argument for why the equation is an identity Explain Mai started by using the distributive property the expression on the left side She then arranged the result to look the same a the right side Since she can write both sides the same way Mai thinks the equation is an identit

Find all integer a such that 2 37 mod 7 20 pts

7 Solve algebraically for x 3600 1 02x 2000 1 04x

10 2 2 2 2 2 marks True or false If true prove it and if false give a counter example i The vector n 2 3 is perpendicular to the line 2x 3y 7 ii The set W A E Man det A 0 is a subspace of the vector space Mnn the set of all n x n matrices with the usual addition and scalar multiplication iii If E and E2 are elementary matrices then the product E E2 is an elementary matrix iy If A is an invertible symmetric matrix then adj A is also a symmetric matrix v The zero vector of any vector space is unique

18 points Let be the vector space of polynomials of degree 2 or less Consider the following two ordered bases of 2 x x 2 2x x 1 x 2 x x 2 x 5 x 2x id a Find the change of basis matrix from the basis to the basis id b Find the change of basis matrix from the basis to the basis

Let W be the set of all vectors 8 x y x y with x and y real Find a basis of W

1 2 l 1 3 3 1 3 marks Consider the system of equations k x y z w 2x 3y 4z w 10 3x 4y 5z 2w 14 2x 5y lw For which values of k and I does the system have i no solution ii a unique solution iii exactly two solutions iv infinitely many solutions K 10 14 3 2R Rz 3R R3 2 R R4 09 D 1 03 I N N H 3 2 www 3 2 1 fund e 2 10 2k 14 3k 3 2 k

How does the graph of f x x 2 4 6 compare to the parent function g x x

a Solve the equation 4x 1 6 1 Show work

or the function f x x 4x 45 solve f x 20 elect one a All real numbers b 9 5 Oc No solution Od 9 5 00

Express the following invertible matrix A as a product of elementary matrices You can resize a matrix when appropriate by clicking and dragging the bottom right corner of the matrix A 0 2 1 2 Number of Matrices 1 Jagal

1 point Find the missing coordinates such that the three vectors form an Orthonormal basis for R 0 6 0 8 1 0 6

2x 5y z 8 b x 2y 3z 13 x 3y z 5 B

Compute an SVD of the matrix A 34 12 6 8 6 8 2 36 36 2 8 6 8 6 12 34 What is an SVD of the matrix UEVT Round to two decimal places as needed

1 Let w be a vector with initial point 4 4 and terminal point 2 5 Write w as a linear combination of the standard unit vectors i and Ow 6i 9j Ow 8i 3j Ow 0i 3j Ow i 6j w 2i 9j

Find the domain and range of the relation and determine whether it is a function What is the domain of the relation OA 1 x 0 OB 1 x 1 OC 6 x 6 3 G E

5 Describe the behavior of the function below as x approaches f x 3 cotx Of 1 as x O f as x Of 1 as x 1 O f 0 as x n O f o as x

17 4 4 25 17 4 Ston 17 4 At least one of the answers above is NOT correct Step 1 Find the x coordinate of the vertex correct incorrect Find the vertex of y x 5x 2 using the vertex formula This problem will not accept decimal answers Please use fractions only Your answer isn t a point it looks like a real number

Use the simplex method to solve the linear programming problem Minimize w 4y 2y2 subject to 3y 2y 60 4y1 y2 40 y 0 y 20 O A 80 when y 2 and y 8 OB 66 when y 0 and y 0 OC 64 when y 4 and y 24 D 48 when y 24 and y 1

8 Find an algebraic expression that is equivalent to the expression tan arccos

etermine the values of a for which the following system of linear equations has no solutions a unique solution or infinitely many solutions You can select always ever a or a then specify a value or comma separated list of values X1 2x2 2x3 1 3x7 6x2 9x3 12 2x7 ax2 7x3 12 Ho Solutions Unique Solution nfinitely Many Solutions Always Always Always Always Never When a SUBMIT AND MARK SAVE AND CLO

58 Find an eigenpair for the matrix 2 3 5 4 2

r is the growth or decay rate It is positive for growth and negative for decay Growth and decay problems can deal with money interest compounded continuously bacteria growth radioactive decay population growth etc So A t can represent any of these depending on the problem Practice The growth of a certain bacteria population can be modeled by the function A t 350e0 0796t where A t is the number of bacteria and t represents the time in minutes a What is the initial number of bacteria 350 round to the nearest whole number of bacteria b What is the number of bacteria after 20 minutes 1720 round to the nearest whole number of bacteria c How long will it take for the number of bacteria to double least 3 decimal places your answer must be accurate to at

19 Find the matrix A if 2 26 10 27 41434 if 4 e 10 26 13 5 b 7 26 3 10

16 Let V v2 3 and v3 Note that V V2 V3 is an orthonormal set Find ca so that C V1 C2V2 C3V3 a b c d c

21 Suppose that A is a matrix with eigenvalues A 9 and 8 with corresponding and 5 respectively Find a22 i e find the entry in row 2 column 2 of eigenvectors the matrix A a 4 b 5 c 1 d 3 e 2

18 Which of the following statements are always true 1 If A is a square matrix then tr A tr A ii If A and B are matrices such that AB B then A I ii If A and B are symmetric matrices of the same size then the matrix AB is also symmetric a 1 only b all of them c i and iii only d i and ii only e ii and iii only

8 Which of the following sets of vectors span R2 i 1 2 1 0 ii 1 1 2 2 iii 1 0 0 1 3 4 a i only b i and iii only c iii only d all of them e i and ii

10 Let V be the subset of R consisting of all vectors of the form a b c d where d a b c Find a basis for V a 1 0 0 0 0 1 0 0 0 0 1 0 b 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 c 1 1 1 3 d 1 1 1 3 1 0 1 2 e 1 0 0 1 0 1 0 1 0 0 1 1

2 Find the Hill cipher of the word AT using the following enciphering matrix 3 A B C D E F G H I J K L M N O P Q R S T U V W X Y 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 a OP b BC c BR d OX e ZP

Find a basis for the null space of A 80A D 0 48840 e a d 1 2 4 5 A 0 1 3 2 1 0 0 1 2 b c

1 A biologist wants to know the width w of a river so that instruments for studying the pollutants in the water can be set properly From point A th biologist walks downstream x 150 feet and sights to point C see figure From this sighting it is determined that 0 54 How wide is the river 0 54 AXI

Let W be the subspace spanned by u and u and write y as the sum of a vector in W and a vector orthogonal to W 7 U 1 4 3 5 The sum is y z where y is in W and z Simplif SIDD is orthogonal to W

L Simplify the following expressions and express your answers with positive exponents a x x 5 d 13 8 JAL M Before we can apply the rules of exponents to expressions in the form of a n where the exponent is a fraction we need to consider what such an expression means Simplify the following expressions a x1 2 x 1 2 c m 2 m 2 9 then m d h 2 h 2 If h The expression a called a b a xxx h1 2 Simplify the following expressions 1 2 is often written as If h 5 5 6 d h 3 h 3 h1 3 1 2 c 5 5 5 5 then x h1 3 1 3 C 1 2 16 broc com what s another way to write h 23 x 2 and is called the ola 1 3 The nth Root If a is a real number and n is a positive integer a a b 41 2 41 2 what s another way to write h 32 b 1 3 27 the nth root of a e v 2 w5 v of a The symbol 8 is

Solve the following system of equations If there is a unique solution enter it as an ordered pair in the form x y If there is no solution enter Inconsistent If there are infinitely many solutions enter your answer as a line in the form y mx b 2x 3y 3 6x9y 9

By hand solve the system using matrices and row operations List each row operation as we have done in class i e R3 10r2 73 Give exact answers in fraction form If the system has no solution state this If the system has an infinite number of solutions give one solution x y z 8 2x 3y z 2 3x 2y9z 9

6 Given a 10 b 6 and c 14 use the Law of Cosines to solve the triangle for the value of B Round your answer to two decimal places 0 60 33 0 80 44 38 21 120 00

7 Use the figure below to find the exact value of the given trigonometric expression 0 sin 2 24 figura not 10 accari to sagla

Rewrite the system as an augmented matrix then use your calculator to write the matrix 10 in row echelon form REF Use back substitution to solve the system Give exact answers in fraction form If the system has no solution state this If the system has an infinite number of solutions give one solution x 5y z 1 3x 11y4z 1 3x 10y 7z 5

3 Evaluate the following expression 4 O 0 4 3 sec y 3 cot 7 y 2 3 3

Find the eigenvalues of the symmetric matrix Enter your answers as a comma separated list Enter your answers from smallest to largest 044 404 444 A1 For each eigenvalue find the dimension of the corresponding eigenspace Enter your answers as a comma separated list dim x

Let f x 3x 4 What is the domain of f x What is the range of f x Determine f x f x What is the domain of f x What is the range of f x

The expenses and income of an individual are given in table form to the right Find the net monthly cash flow it may be negative or positive Assume that amounts shown for salaries and wages are after taxes and that 1 month 4 weeks Income Part time job 1100 month Student loans 8400 year Scholarship 6500 year The net monthly cash flow is Round to the nearest dollar as needed Rent 550 month Expenses Groceries 50 week Tuition and fees 6500 year Health insurance 25 month Entertainment 100 month Phone 65 month

16 14 The solution is 7

Click here to watch the video 17x 9 Find the quotient 3 cis 17x 9 3 cis 9 cis 2x 9 in rectangular form using exact values 2x 9 cis 9 Simplify your answer including any radicals Use integers or fractions for any numbers in the expression Type your answer in the form a bi

6 From a point 125 m from the foot of a building the angles of elev the building s flagpole are 40 and 38 respectively 380 409 125 m Calculate a the height of the building the Height of the building is 236 66m Tall Pe 125 38 125 x x 138 125 125 7 125 78m T H 164 88 125 78 b the length of the flagpole 230 66 104 88 O X x Tanus Tan 88 125 78 x 104 88m 125 Tan 38 buo a Tan38 125 78 Tan 40 X Tan Uc

A garden is in the shape of a rectangle 43 feet long and 13 feet wide If fencing costs 6 25 a foot what will it cost to place fencing around the garden 118 25 3 493 75 350 00 700 00