Matrices & Determinants Questions and Answers

Ralph Chase plans to sell a piece of property for 145000 He wants the money to be paid off in two ways a short term note at 11 interest and P Note Interest Time Amount Rate in years 1 1 X 11 a long term note at 8 interest Find 145000 x 8 the amount of each note if the total annual interest paid is 14000 Interest Paid x 0 11 1 145000 x 0 08 1 The amount of the 11 note is and the amount of the 8 note is

Find the roots of the polynomial p x x 9x 10 O 10 and 1 O 1 and 10

0 Let A P 3 Find a nonzero vector u in Nul A u 1 7 1 1 16

2 Is the set of vectors 2 3 1 4 1 1 0 7 1 a basis for R

5 Find a basis for the solution space of the following homogeneous linear system and find the dimension of that space 3x1 x2 x3 X4 0 5x1 X2 X3 X4 0

1 Are the following sets of vectors linearly independent Why a u 2 3 and u 1 2 and 3 3 1 in R b p 1 2x x and P 2 4x 2 x in P2 c A 1 03 05 in M 2 and B

6 Find the basis for the given subspaces of R and state the dimension a 2x 3y 4z 0 b 2x 3y 0

In planning her retirement Liza deposits some money at 2 interest with twice as much deposited at 2 5 Find the amount deposited at each rate if the total annual interest income is 700 She deposited at 2 and at 2 5

Find the slope of the secant to the curve f x x 5 between x 2 and x 1 O a 3 b 1 3 c 3 d Submitting Your Test

Determine the composite function fog and gof For each composite function state the domain and range a f x 5 2x x and g x 1 6x b f x 3 and g x log x c f x sin x and g x 4x 6x 2 d f x 4x 1 and g x x

e differential equation y ay by cy dy 0 s a solution y 2te2 6e t sin 3t for real numbers a b c and d hen

A ship is sailing due north At a certain point the bearing of a lighthouse 8 3 km away is N35 5 E Later on the captain notices that the bearing of the lighthouse has become 344 E How far did the ship travel between the two observations of the lighthouse The ship travelled km between the two observations Do not round until the final anewer Then round to the nearest tenth as needed

1 2 3 1 0 1 X 3 2 y 2 Y 2 3 y 2 a Y 24 19 y 2 y 21 y 2 y 2 200 exponen y they show exponential growth or decay x y 9 8 7 10 S 6 5 4 3 2 3 21 Or 11 2 31 1 2 3 4 5 6 7 8 9 6

Which angles are supplementary to each other 25 and 24 2 1 35 4 22 and 21 24 and 23 21 and 24

3 Below is a Venn diagram illustrating the primary language spoken in the home at one of our local high schools all answers are to be to the nearest 10th of a percent Spanish 33 3 8 35 Chinese 13 English a 3pts How many students participated in the survey b 3pts Choosing a home at random what is the probability they speak only in the household English c 3pts Choosing a home at random what is the probability they speak Spanish in the household d 3pts Choosing a home at random what is the probability the speak English and Chinese e 3pts Choosing a home at random what is the probability th speak Chinese and English and Spanish in the household

Determine if the columns of the matrixx form a linearly independent set 13 3 0 3 10 7 2 29 2 8 Select the correct choice below and if necessary Mil in the answer box es within your choice OA The columns of the matrix do not form a linearly independent set because there are more entries in each vector Type whole numbers man there are vectors in the set OB The columns of the matrix do not form a linearly independent set because the set contains more vectors than there are entries in each vector Type whole numbers OC Let A be the given matrox Then the columns of the matrix form a linearly independent set since the vector equation Ax 0 has only the trivial solution OD The columns of the matrix form a linearly independent set because at least one vector in the set is a constant multiple of another

Graph the image of AVWX after the following sequence of transformations Translation 8 units right and 16 units up Rotation 90 counterclockwise around the origin 14 12 10 8 6 4 2 14 1 2 10 8 6 4 2 X 2 V 4 6 8 1 0 1 2 1 4 N 2 4 6 W ES 8 10

Which two triangles are congruent by the SAS Theorem Complete the congruence statement F J HHH E H H D I S 1 R HHH

Let A 3 1 4 9 k For A to have 0 as an eigenvalue k must be

a Use the cross multiplication method to calculate the determinant of matrix A b Use the cross multiplication method to calculate the determinant of matrix B c Use the co factor expansion method to calculate the determinant of matrix C 2 3 2 31 3 2 2 20 3 3 0 2 One mark will be taken off for not choosing the best row or column to perform your calculation for part c Note

8 Let T R R be a linear transformation Find 3 a J d 76 G 3 and

1 1 We put V v1 2 and W 3 Let t and w a Compute an R basis of Vnw b Compute an R basis of V W W be vectors in R

nd the rank and nulity of the matrix A 0 1 1 2 1 1 1 0 2 with entries in ZZ3 Rank number of leading ones in its RREF Nulity number of columns without leading ones in its RREF

Consider the following piecewise defined function 2 x 5 5 if x 5 36 5 if x 5 X 9 Step 1 of 3 Evaluate this function at x 10 Write the exact answer Do not round If the answer is undefined write Und as your answer Answer 2 Points f x f 10 Keypa Keyboard Shorte

2w 2w w 6 54 4 v 2x 2 y 42 486 32 5y z 2y 4z AnswerHow to enter your answer opens in new window 2 Points Selecting a checkbox will replace the entered answer value s with the checkbox value If the checkbox is not selected the entered an 11 y Z Not Possible

3 Solve the following system of linear equations using the inverse of the coefficient matrix Use the inversion algorithm to find the inverse x 8z 1 2x 4y 6z 1 2x 5y 3z 2 035

4 Find the determinant of the following matrix using elementary row operations 4 2 2 B 2 2 6 3

11 Find the volume of the parallelepiped made by u 3 2 6 v 1 2 7 and w 2 0 2

6 Let u 3 2 6 and a 1 2 7 find the vector component of u along a and the vector component of u orthogonal to a

10 Find the area of the triangle determined by points A 2 2 0 B 0 4 3 and C 1 0 2

10 range of this exponential function 8 6 4 yly 3 all real numbers yly 3 yly 3 yly 3 2 104 8 6 4 2 0 2 4 6 8 10 2 4 6 8 X 10 A

Let A 0 0 0 where r is a real number Answer the following questions 0 1 Find all eigenvalues of A and the corresponding eigenvectors 2 Find an expression for A for any positive integer n 3 Find an expression for Q LA12 Please help me with a solution to Question 3

Determine whether the set of vectors in M22 is linearly independent or linearly depender 2 3 S C JE 14 9 DE S

Dawson wants to pay a 500 credit note that is due in 1 year His creditor is will ing to accept early payment of the loan by discounting it at 3 5 annual interest com pounded monthly How much can Dawson pay today to eliminate his debt

then A 1 6 00 HERE A 07 0 0 0 2

We find the domain by finding all real inputs Step 2 The domain includes all real numbers except inputs x values that result in real outputs 1 values that result in a denominator of Select and 2 values that result in an even root of Select C outputs y values For the function y x 8 there are no such restrictions on the domain When there are no restrictions the domain of a function consists of all real numbers because there are no exclusions The inputs come from all values on the real number line which span from negative infinity to positive infinity Enter the domain of the function y x 8 using interval notation

y t y t te y 0 2 y 0 1 by laplace

Given is a equilateral triangle What colours do you need to specify for the vertices in order to make the centre point light grey RGB value 2 3 2 3 2 3 Select one O a The vertices must be cyan red and yellow O b The vertices must be red green and blue O c O d The vertices must be red yellow and blue O e The vertices must be cyan magenta and yellow The vertices must be magenta green and blue

Consider the point P represented in 3D homogenous coordinates as P What statement about p the transformed point P after first performing a translation by O a O d P O b None of the others O c O e P P 2 P COON CON 3231 Toor TTO then a rotation by 90 degrees around the z axis then a scaling by 2 in the x and y axes is tru

c Find the value of the following determinant 1 2 8 5 T 7 1 3 9

5 Let G Z9 X Z12 X Z16 a Find all subgroups of G of order 144 b Let H 3 3 6 the cyclic subgroup of G generated by 3 3 6 Determine G c Determine the order of the element 4 5 2 H in the factor group G H

Solve log x 3 logg 3x 5 0 Select the correct answer below 01 0 1 05

Given the diagram below U 1 5 A B V 9 2 answer the following questions W Which of the vectors A B or C also represents the vector u Choose B Which of the vectors A B or C also represents the vector v C Which of the vectors A B or C represents the resultant vector u v A none of thes

Let G be the solid in the first octant that is inside cylinder x y 4 outside the 2 sphere x y z 2 4 and between the planes y 0 and y x 3 1 Find the volume of G using a triple integral in spherical coordinates 2 Set up do not evaluate an iterated triple integral in cylindrical coordinates equal to mass of G if the density at any point x y z 2 in G is 6 x y z x 1

A chemical manufacturer wants to lease a fleet of 28 railroad tank cars with a combined carrying capacity of 420 000 gallons Tank cars with three different carrying capacities are available 6 000 gallons 12 000 gallons and 24 000 gallons How many of each type of tank car should be leased Let x be the number of cars with a 6 000 gallon capacity x be the number of cars with a 12 000 gallon capacity and x3 be the number of cars with a 24 000 gallon capacity Select the correct choice below and fill in the answer boxes within your choice OA The unique solution is x and x3 Simplify your answers B There are multiple possible combinations of how the tank cars should be leased The combinations are obtained from the equations X 2t 49 x 3t 77 and x3 t forsts Simplify your answers Type integers or simplified fractions OC There is no solution X

In Exercises 33 36 write each matrix equation as a system of linear equations without matrices 33 2 3 30 1 34 13 90 9 3 kontroll 2 0 1 0 1 1 0 35 03 1 0 1 36 0 1 0 0 1 1 37 x 5 10 0 y 6 In Exercises 37 42 AX B a Write each linear system as a matrix equation in the form b Solve the system using the inverse that is given for the coefficient matrix The inverse of 266 2 7 277 2x 6y 6z 8 2x 7y 6z 10 2x 7y 7z 9 6 is

What is the common ratio of the sequence 1 1 3 4 2 3 3 2 4 3 3 4 2

Exercise 7 2 6 Find A4 for the matrix 1 0 0 1 1 02 002 A 0 000 1 0 of Exercise 7 1 8 using the eigenvalues and eigenvectors of A

Give an example of matrices A B M2x2 such that null A B is not a subset of either null A or null B

Which one of the following is a subspace of M2x2 20 a b a d a b 8 1 a b a b a b 0 c d 1 d