Matrices & Determinants Questions and Answers

Change from degrees to radians 210 7

10 Apply the singular value decomposition to show that A Rmxn with rank r can be written a A BC where Be Rmx CE Rx and both B and C have rank r

If 900 are deposited into an account with 5 5 interest rate compounded quarterly what is the balance after 20 years F Round to the nearest cent Enter

1 a Use the coefficient matrix to classify the stability of the origin the only equilibrium point for the following linear system x x y y 3x 4y

Let v 3 1 8 5 3 and p 4 Determine if p is in Col A where A V V V3 Is p in Col A OA Yes because the augmented matrix A p is consistent OB No because the vectors V V and v are linearly dependent OC No because the augmented matrix A p is not consistent D Yes because the vectors v V and v are linearly independent CHILD

8 You re a tactical officer on a space ship The ship s computers detect an unidentified space vessel in the shape of a cube and you need to get an idea of the size mass of the cube so you can assess the risk Sensors indicate each side of the cube is s 0 000000085 in Given the formula V 8 find the volume of the cube whose side is s 0 000000085 in V in

3 Since birth Hachiro has had a savings account that has been growing at a rate of 4 per year The amount of money in the account can be modeled by the equation y P 1 04 where y is the value of the account and x is the number of years and P was the original deposit amount How much money is in the account if Hachiro is 32 years old and the original deposit was 8000 O 28037 79 O 28582 52 O 28064 47 O 27188 45 None of these is correct

4 Evaluate the expression if t 7 and u t u answer x

7 A newly hatched snake weighs 1 0 lb This weight increases by 3 percent each day for the next 3 weeks of its life Write an exponential equation that models the growth of the snake and approximate the weight of the snake after 3 weeks Round to the nearest hundredth y lbs

1 Use your calculator to find the value of y if x 4 Round to the nearest hundredth y 765 1 39 y

All vectors are in Rn Check the true statements below A If a set S up has the property that u U j then S is an orthonormal set 0 whenever i B The orthogonal projection of y onto v is the same as the orthogonal projection of y onto cv whenever c 0 C An orthogonal matrix is invertible D Not every orthogonal set in R is a linearly independent set E If the columns of an m x n matrix A are orthonormal then the linear Ar preserves lengths manning r

2 Solve the system of equations x y z 3 x 1 y z 2 2 y

4 What does y equal in the solution of the system of equations below 2y8x3z 66 21x 0 2z 2x 24 O 02 05

The dot product of two vectors can be written in the form x s 11 I1 I2 t in R is defined by a y x1y1 242 Inyn The vectors and y are called perpendicular x 0 Any vector in R perpendicular to and y 3 A Y1 Y2

11 State the number of solutions without graphing 11y 55 2x y x 5 O One solution O No solutions O Many solutions Two solutions

4 On a separate sheet solve the system of equations by graphing Check your answer with the feedback 6x 12y 36 y x 1 x y

Problem 8 Find a least square solution for the equation 90 0 3 2

How long will a package containing a dozen gauze pads last if a dressing that requires one pad is changed four times a day A B C 2 days 3 days 4 days 5 days

1 a Solve the following stem of linear equations by LU Method 15 pts b Solve the following system of linear equations by Cholesky Method 15 pts c Solve the following system of linear equations by Gauss Elimination Method 20 pts 2x x 3x 5 X 3x 2x 11

6 Find the image of the line y 2x 3 under the transformation by matrix 2 A 5 3 m

7 Find the image of a triangle with three vertices 0 0 1 1 and 1 2 under the transformation by matrix A 23 Draw both triangles in the x y plane

02 20 20 Problem 6 5 points Find the line that best fits the data M A A

2 By inspection of matrix R obtained for matrix A find the rank of A the nullity of A the basis for row space the basis for column space of matrix A A 0 1 2 2 2 2 0 1 3 1 1 3 3 1 2 R 11 1 0 0 0 0 1 II 0 0 0 0 0 1 0

2x 2x 2x3 4 2X2 2 x 5x 2x3 1 8 x x 4x 11

Dilate the given triangle with a magnitude of 3 6 3 3 1 3 9 9 31 31 CO

Reflect the given triangle over the y axis 3 3 3 6 31 3 T 3 6 3 0 Enter

sin x dx Your ID s last three digits are abc Let m 1 a b c Apply Simpson s Rule with n 2 to solve the integral Just type the answer with 2 decimals Your answer should be h stepsize value and the result of Simpson s rule Warning Below the formula is only for one interval Inside sin there are no degrees but radial numbers Simpson s Rule Sn Ax 3 f xo 4f x1 f x

A 5 Find the cigenvalues and the eigenvectors of A Is A diagonalizable Explain If so find an invertible matrix P and a dingonalizable matrix D such that PAP D Write A as a product of P a power of D and P

The length of two objects are in the ratio of 2 1 If each were 3 inches shorter the ratio would be 3 1 The longer object is

Compute the square root of the matrix A where A 8 10 3 10

Let A be an 10x10 matrix define T s A sl Show values such T s is invertible every diagnol entry of T s 1 is of form p s q s where p s and qs polynomials in s What are thedegrees of ps qs expalin

t p p 1 p 0 p 0 p 1 Let t p3 s2 be the linear transformation Fi

Find the matrix P if it exists which diagonalize the matrix 1 3 3 3 5 3 6 6 4

Let A nxn matrix such A 2 A Show that if delta is eigenvalue of A then delta 0 or 1

Identify the key features of the graph of h x Select all that apply 1 0 8 5 4 2 10 8 6 4 2 4 6 co 8 10 2 1 9 8 10 This image cannot be easily described If you need an explanation of this image please ask your teach help The function has an x intercept of 0 1 and no y intercepts As x approaches positive infinity h x approaches negative infinity As x approaches positive infinity h x approaches 0 which means that the function has an asymptote of y The function has a y intercept of 0 1 and no x intercepts As x approaches negative infinity h x approaches positive infinity O The value of the function increases throughout its domain

1 1 v1 0 v2 1 1 0 Find an orthogonal basis u1 u2 for W spar v1 v2 C

a 203 9 1 9 0 0 1 Are therean invertible 3x3 matrix P and a diagonal3x3 matrix D such that P 1 AP D no explain if yes find P D

NSS 9 5 45 Two tanks each holding 50 L of liquid are interconnected by pipes with liquid flowing from tank A into tank B at a rate of 4 L min and from tank B into tank A at a rate of 1 L min The liquid inside each tank is kept well stirred Pure water flows into tank A at a rate of 3 L min and the solution flows out of tank B at 3 L min If initially tank A contains 2 5 kg of salt and tank B contains no salt only water determine the mass of salt in each tank at time t 0 3 L min Pure water A x 1 50 L x 0 2 5 kg 4 L min 1 L min B x 1 50 L 0 0 kg 3 L min

Nevaeh is solving this absolute value inequality using a graph x 8 7 She graphs the following in Desmos y x 8 y 7 Here is her graph 5 O 7 x 8 x 10 This image cannot be easily described If you need an explanation of this image please ask your teacher help Using the graph what is the solution to the absolute value inequality O No solution 5 0

O g x x 2 O g x x 2 Og x x 2 O g x 4x 2 Question 2 Describe the transformation from f x x to the following function f x x 4 O Translation Right 4 Translation Up 4

Match the domain and range for the graph below 1 0 The domain is 5 4 2 10 CO 6 at 4 2 2 4 6 CO 10 2 4 9 8 10 0 infinity

5 Let A 1 1 1 3 and b NN 2 1 1 Find the least squares solution of the system Ax b and find projR A b 2 Find an orthonormal basis for the column space of and find projR A b

For the system represented by the state equation X x Ax t the state response x t is given by x t Determine the system matrix A at at when x 0 1 B

Determine the domain of the following graph 12 11 10 9 8 7 6 5 4 3 2 1 4 5 c c i i c n m 2 6 7 8 9 10 H 11 10 2H998765 432 12 11 Y 1 2 3 4 5 6 7 8 9 10 11 12

Let A 2 3 7 5 and u 7 Define a transformation T R R by T x Ax Find T u the image of u under the transformation T

Question 11 wo Find the standard matrix of the linear transformation T T R R rotates points about the origin through angle T 4 radians with counterclockwise rotation for a positive

You should be able to find this without a calculator cos cos 0 3

T 0 0 1 sin cos sin 0 0 cos 0 0

5 S 15 Let T T2 are linear transforms in R2 defined by T x x 2x2 2x1 x2 T x 3x x2 x1 3x2 nd a standard matrix for T 2T1

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