Matrices & Determinants Questions and Answers

1 7 12 Which function describes a parabola that opens downward and has a vertex of 2 3 a f x x 2 3 b f x x 2 3 c f x x 2 3 d f x x 1 2 7 d f x x 2 3 13 List the sequences of steps required to graph the function f x x 4 6 a horizontal translation 4 units to the right vertical stretch by a factor of 1 and a translation 6 units down b horizontal translation 4 units to the right reflection over the x axis and a vertical translation 6 units down c horizontal translation 4 units to the left vertical translation 6 units up and a reflection over the x axis d horizontal translation 4 units to the left reflection over the x axis and a vertical translation 6 units down 14 List the sequences of steps required to graph the function f x 3 x 7 1 a horizontal translation 7 units to the right vertical stretch by a factor of 1 and a translation 6 units down b horizontal translation 7 units to the right vertical stretch by a factor of 3 and a translation 1 unit down c horizontal translation 7 units to the left vertical stretch by a factor of 3 and a translation 1 unit up d horizontal translation 7 units left reflection over the x axis by a factor of 3 and a translation 1 unit down 15 Consider a parabola P that opens upward and has a vertex 1 3 Now find the equation of a new parabole that results if P is reflected over the x axis and translated 3 units down a f x x 4 3 b f x x 1 6

10 2x 9 58 115 3x 5 9 X zolgnA vist 12 qmo be algna dous Tomad 05 X 351 001 X 30 081 201 PF 11 A x 16 63 1010

Find the midpoint M of the line segment joining the points A 4 1 and B 8 5 B M 0 pla

Kyle works at a donut factory where a 10 oz cup of coffee costs 95 a 14 oz cup costs 1 15 and a 20 oz cup costs 1 50 During one busy period Kyle served 17 cups of coffee using 244 ounces of coffee while collecting a total of 20 10 How many cups of each size did Kyle fill Kyle filled 10 oz cup s 14 oz cup s and 20 oz cup s

25 4 points Melissa s dinner serves desserts One serving of ice cream and 2 servings of apple pie provides 800 calories 3 servings of ice cream and 2 servings of apple pie provides 1280 calories Set up a system of equations define your two variables and use either method you choose to determine how many calories are in one serving of ice cream

9 2a 2 3a 5a 6 A 28a 63a 19a 20 B 6a 16a 22a 12 C 10a 27a 32a 14 D 56a 29a 8a 20 10 10 8n 7 7n 5n 3 A 12n 46n 14m 42 B 8n 32n 8n 48 C 15n 17n 22n 24 D 56n 89n 59n 21

Factor the polynomial by grouping 9x 9x x 1 A x 1 9x 1 B C x 1 9x 1 x 1 9x 1 D x 1 9x 1

How does the graph of g x x 2 3 6 compare to the parent function of f x x g x is shifted 2 units to the right and 6 units down g x is shifted 2 units to the right and 6 units up O g x is shifted 2 units to the left and 6 units down O g x is shifted 6 units to the left and 2 units down

The payoff matrix and strategies P and Q for the row and column players respectively are given Find the expected payoff E of each game Round your answer to two decimal places 0 2 2 3 1 4 1 4 1 2 E 92 P 4 42 4 Q

Make a scatter plot of the points in the table type of function from those below that would best represent the data X 5 3 2 5 6 8 10 Y1 65 23 12 15 32 78 140 The data is best represented by a quadratic function The data is best represented by an exponential function The data is best represented by a linear fung

2 Determine if each of the following is a group a monoid that is not a group a semigroup that is not a monoid or not a semigroup Explain your answer for each one Z addition on the set of positive integers It farms bernise a group It satisfies closure associativity identity element and every element has an inverse ii Q subtraction on the set of rational numbers This is not a semigroup because the set of rational numbers under subtration isn t closed under subtraction iii C32 addition on a 32 hour clock It satisfies Closure associativity identity element and every element has an in verse It forms a group iv C15 0 multiplication on a 15 hour clock with 0 removed This doesn t form a group or monoid Not element has an inverse every v 0 0 1 0 1 U the set is P 0 1 and the operation is union a semigroup but not a monoid This is on the power set of 0 1 forms a sem igroup The set is closed under the union operation which is associative It doesn t have an identity element so it s not a mencid vi 0 1 2 where x y y for example 0 2 2

1 3 Multiply Rational Expressions Simplify each expression 35m 65n 10 28n 4n 1 A 10n n 1 3 n 3 n 1 3 C 21n 53n 8 6n 13n 8 A 2 5n 3 C70 B Sn B D 3n 50n 80 n 1 n 9 3 An n 8 20n 10n 42n 6 9 5n 9 n 2 2 4 his reserved 2 7r 70 5r 39r 54 35r 42 10r A 4 C r 1 r 8 7 6 r 3 B v 3 A 7 C v 9 v 4 6v 35v 5 63y 9v 5v 45 r 10 r 9 10r D r 5 2 3 v 9 D 7 v 6 B

x 3y 3z 3 2yz 0 4yz 4x 3 a Express the system in matrix form A 6 and indicate each matrix vector below A m below 18 b Use technology to find the inverse of the coefficient matrix c Solve the system by computing A 6 without technology Enter your answer as an ordered triplet

If it exists find the inverse of the matrix A If it does not exist enter DNE 10 5 4 5 3 01 5 A

Consider the matrices below AB A BA 1 7 2 8 3 3 3 B If possible find each product below If not possible enter none 7 6 49 4 7

The matrix B has dimensions m x n a Identify the value of m b Identify the value of n c Identify the value of b12 B d Identify the value of B 22 13 13 1 14 13 12 16 2

Find QR R Q 5 S Write your answer as an integer or as a decimal rounded to the nearest tenth QR

What is the y intercept of the function f x 2 3 Oy 2 Oy 3 Oy 4 Oy 5

6 Solve the system of equations 3x 2y z 4 10 x 3y 4z 7 a 2x y 2z b

9 Determine whether the linear operator T R R defined by the equations is one to one if so find the standard matrix for the inverse operator and find T w w wz 10pts W x 2x Iz W 2x x 4x3 W3 7x 4x2 5x3

a fog x and its domain b gof x and its domain a fog x 5x 3 Simplify your answer The domain is 3 5 0 Type your answer in interval notation b gof x 5 x 3 Simplify your answer The domain is Type your answer in interval notation

You have decided to plant a vegetable garden containing tomatoes peppers and cucumbers The garden will have 61 plants There will be twice as many tomato plants as pepper plants Tomato plants cost 2 00 each Pepper plants cost 5 00 each Cucumber plants cost 5 00 each You have 197 00 to spend on the plants Create a system of equations and the corresponding augmented matrix Then use technology to put that matrix in reduced row echelon form Then give the number of plants of each type that should be purchased Tomatoes Peppers

Exercise Show that the Hademaard gate H below is a unitary operator Describe what it does to 10 and 1 H 4 2

Problem 9 a A Problem 9 b 11 1 A 0 11 0 B 1 0 0 0 0 1 0 0 C 1 11 1 0 11 0 D 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 E 1 11 1 0 11 0 1 10 0 F 1 11 1 0 11 0 1 10 0 1 1 1 G 1 1 1 H 1 11 1 Select 0 1 11 14 0 1 a Find a basis for the column space of A b Find a basis for the nullspace of A 1 1 1 Select A 1 2 1 0 1 7 0 1 B 1 1 0 1 0 1 C 1 0 1 D 1 E 1 0 1 F 1 0 1 G 1 1 0 1 0 1 1 1 0 1 0 1 0 1

2 We can already study our first gate at this point Consider the NOT gate 0 G 1 N 2 1 Note that this matrix is unitary in that N NT N where the symbol means take the transpose the complex coniugate

Problem 7 Find a basis for the subspace of R4 consisting of all vectors of the form a b c d where c a 7b and d a 5b A 0 1 7 1 1 1 0 5 B 1 1 0 1 0 1 7 5 C 1 0 7 1 0 1 1 5 D 1 0 1 1 0 1 7 5 E 1 0 1 1 1 0 7 5 F 0 1 7 1 0 1 1 5 G 1 0 7 1 1 0 1 5 H 1 1 0 1 1 0 7 5

Use the Quotient and Reciprocal Identities to match equivalent expressions sin 0 1 1 1 1 1 1 cos 0 a cos 0 csc sin 0 sec 0 cot e cos e tan e sin 0 sin e b cos 0 c tan e d csc 0 e sec 0 f cot 0 1

4 Determine the slope and y intercept of this graph a b 2 0 3 slope y intercept 1 5 slope 1 5 y intercept 3 4 C d 3 slope y intercept 1 5 3 slope y intercept 1 5

x 1 Rewrite this equation in general form 3x 5 3x y 3x 5

10 2 0 2 1 5 3 1 5 4 1 1 Not equivalent a Multiply R row 1 by 2 Enter the new matrix 10 2 0 2 1 5 3 1 8 11 10 2 5 1 3 Not equivalent 1 5 4 3 1 1 1 1 5 3 1 10 202 b Now replace R3 by adding 3R2 to R3 Enter the new matrix c Now switch R and R Enter the new matrix 1150 254 10 2 0 5 10 2 1 5 8 11 1 5 10 2 Preview 3 1 277 1 1 30 Preview 2 0 3 1 10 2 Preview 1 2

Divide 4 x 6x 15x 28 x 4 3 x 4 A x 2x 8 B x 2x 7 C x 2x 6 D x 2x 8 5 x 4 5 x 4

Enter the augmented matrix for the following system of equations Note the following You do not need to solve the system You do not need to insert a vertical line in the position of the equal signs Do not reorder the equations Do not multiply any equation by some non zero constant other than one The variables should be left of each equal sign and the constant to the right Order the first three columns of the augmented matrix as the coefficients of x y and respectively augmented matrix 5z x 3 y 1 y z 2y x 1 3z

For each matrix choose one option Either indicate that the matrix is in reduced row echelon RRE form or indicate the ONE condition that the matrix DOES NOT satisfy Matrix 1 O The matrix is in RRE form O 5 0 4 01 5 0000 1 0 There is a leading entry of a row that is NOT one Leading ones DO NOT appear further to the right from row to row O There is a row of zeros that is NOT at the bottom of the matrix O NOT all other entries in a column with a leading one are zero

Consider the parallelepiped Pin R determined by the vectors u 2 1 1 v 1 3 1 2 and w 1 6 2 8 Use the parallelogram determined by u and v as the base of P a 2 marks Find the area A of the base of P A If needed enter as sqrt z b 1 mark Find the volume V of P c 2 marks Find one vector n orthogonal to the base of P so that the volume of the parallelepiped determined by u v n equals the volume P

0 5 4 3 Amplitude e 9 8 7 16 5 1 24 Show Transcribed Text Close Based on the graph e determine the amplitude midline and period of the function 4

Problem 5 Let W be the subspace of R4 spanned by the vectors u 1 0 1 0 u 0 1 1 0 and u3 0 0 1 1 Use the Gram Schmidt process to transform the basis u U U3 into an orthonormal basis 6 4 v 2 0 2 0 v V6 V6 V6 0 v3 V2 3 B V 2 0 2 0 v2 6 V6 6 0 v3 3 v 3 3 66 13 131313 6 6 2 1 2 0 2 0 v 6 V6 V6 0 v3 V2 3 Problem 5 Select 2 0 12 0 v 6 V6 V6 0 v3 3 V2 3 E v 2 0 2 0 v2 V6 V6 V6 0 v3 D v 2 0 V2 0 v2 V6 V6 V6 0 v3 3 V3 V V V2 3 6 3 3 3 3 6 6 6 F v 2 0 2 0 v2 V6 6 6 0 v3 3 3 3 3 3 6 6 3 3 3 6 6 6 G H v 2 0 2 0 v 6 6 6 0 v3 V2 V6 6 3 3 3 3 3 6 3 V3 3 6

Problem 10 a Problem 10 b A 1 2 5 4 Just Save 1 2 3 0 a Find the dimension of the row space of A b Find the dimension of the nullspace of A Problem 10 Your Answer 3 3 5 6 7 9 1 0 Submit Problem 10 for Grading Attempt 1 10 a 10 h Attempt 2 10 a 10 h Attempt 3 10 a 10 h

Jeremy can run no more than 1 5 miles every 10 minutes At this pace Graph the Inequality to represent how many miles Jeremy can run in 1 hour 1 5x10 15 15x6 90

p exceeds 24 24 38 0 38 25

Problem 8 Let W be the subspace of M12 12 i e 12 x 12 matrices consisting of all matrices whose diagonal entries are zero Find the dimension of W Problem 8

Find an orthogonal matrix A where the first row is a multiple of 0 1 0 A

Problem 7 Find a basis for the subspace of R4 consisting of all vectors of the form a b c d where c a 7b an d a 5b Problem 7 A 0 1 7 1 1 1 0 5 D 1 0 1 1 0 1 7 5 G 1 0 7 1 1 0 1 5 Select B 1 1 0 1 0 1 7 5 C 1 0 7 1 0 1 1 5 F 0 1 7 1 0 1 1 5 E 1 0 1 1 1 0 7 5 H 1 1 0 1 1 0 7 5

roblem 1 Which of the following sets of vectors are independent i 1 2 1 3 2 0 2 4 1 ii 1 2 1 3 2 0 5 6 0 iii 1 2 1 3 2 0 7 10 2 Problem 1 A none of them B ii only C i only D i and iii only E ii and iii only F iii or G all of them H i and ii only F

Solve the given system of equations If the system has no solution say that it is inconsistent x 2y 3z 13 2x y z 1 3x 2y 2z 13 Select the correct choice below and fill in any answer boxes within your choice OA The solution is x y and z Type integers or simplified fractions OB There are infinitely many solutions Using ordered triplets they can be expressed as x y z x y z any real number Simplify your answers Type expressions using z as the variable as needed OC There are infinitely many solutions Using ordered triplets they can be expressed as ou

11 ABC geni gen DBE ye benzer B l m A X in de erini bulmak i in bir denklem kurun ve z n A 4x 3 7x 30 B E

Problem 3 Which of the following sets of polynomials form a basis for P2 P1 1 3x 5x P2 5 x ii P 1 x P2 10x x P3 1 11x x iii P 1 P2 1 6x P3 x P4 x Problem 3 A all of them B i only C i and ii only D iii only E none of them F ii and iii G i and iii only H ii only Select

Problem 2 Let p 8x 6x 3 Find the coordinate vector of p relative to the following basis for P2 P1 1 P2 1 x P3 1 x x Problem 2 Enter the coordinates separated with commas

a Find the center radius form of the equation of the circle with center 3 5 and radius 4 b Graph the circle a The center radius form of the equation of the circle is x 3 y 5 16 Type an equation b Use the graphing tool to graph the circle Click to enlarge graph 4 10 B Ay 10 8 6 4 2 2 8 40 x

Convert the Cartes in coordinate 6 4 to polar coordinates 0 0 2 r Preview

Rationalize the denominator 3 8 2 3 3 5 2 3 8 2 3 3 5 2