Matrices & Determinants Questions and Answers

a Write a set of ordered pairs x y that defines the relation b Write the domain of the relation c Write the range of the relation d Determine if the relation defines y as a function of x Actor x Tom Hanks Jack Nicholson Robert De Niro Number of Oscar Nominations y 8 12 6

5 3 Determine whether each equality det AB det A det B and det A B det A det B holds det AB

Which of the following sets of vectors in M22 the vector space of 2 x 2 matrices with real number entries usual addition and scalar multiplication are orthonormal with respect to the inner product A B trace BT A Select ALL that apply O 1 2 1 2 1 2 1 2 1 2 10 0 0 1 2 1 2 12 01 01 1 1 1

20 Are the matrices A 21 14 023 027 3 3 4 4 0 2 3 4 and B 1 similar

solve the system of Answer equations 3 x2 423 24 11 3x1 x2 5 4 1 x1 3x2 2 a3 54 1 1 3x2 Note Enter your answer as list of vectors as follows if you believe that the general solutions is

Let d ghi Sa 56 10c A d e2f 9 h 21 Then A B C 3 and let B and a 2d b2e c 2 NOTATION h det C

Let A 3 2 7 8 17 1 C22 30 000 5 5 29 0 Then the cofactors C12 2 2 6 3 0 0 0 00 1 the determinant det A and 4 Recall Cramer s Rule Given Ax b if det A 0 then If Ax 14 4 4 0 0 then ag det A det A 1 where A is obtained from A by replacing its ith coll Simplify your answer

Let T R R be the linear operator that is a reflection versus the line y x followed by a rotation clockwise by and angle of 2 followed by a dilation by a factor of 2 Which ONE of the following is the standard matrix for T HINT use T T e T e2 where e e2 is the standard basis for R O N L O 0 O 0 2 2 2 012 1 2 2 0 0 2 0

Let V M2x2 with inner product A B trace BT A 0 3 B 1 3 5 Let A Then A B All answers are numeric 0 2 and B

Let A a A220 1 0 0 b A 220 2 1 0 Given that P diagonalizes A then compute each of the following powers of A 8 0 and P 1 1 0 1 00 4 0 1

6 By applying row operations to produce an upper triangular U compute 2 3 0 6 6 1 0 0 3 020 07 det 1 2 1 and det 2 1 0 0 1 0 0 2 1 0 2 1 2 1 0 1

Determine whether the set of vectors k e In x 2x is linearly independent for any nonzero value of k using the Wronskian 8 marke

28 29 section 11 3 Solve for x Answer 2 X 1 X 6 1 3 0 7 2 section 11 3 Evaluate the following determinant using expansion by minors method leave ans in terms of I j and k

Which of the matrices that follow are in row echelon form ref Which are i reduced row echelon form rref 100 1 30 1 2 3 4 A 6 6 9 B 000 C 001 0012 0 0 1 000 9 9

Let A 10 5 0 4 2 and b 48 3 6 a Is b in a a a O No OYes 6 Denote the columns of A by a a2 a3 and let W Span a a a a Is b in a a a How many vectors are in a a a3 b Is b in W How many vectors are in W c Show that a is in W Hint Row operations are unnecessary

Approved Disapproved A poll of 3128 people revealed that of the respondents that were registered Republicans 827 approved of their representative s job performance 410 did not 735 had no opinion Of the registered Democrats 241 approved of their representative s job performance 435 did not and 76 had no opinion Of those regist as Independents 119 approved 111 did not approve and 22 had no opinion Of the remaining respondents who were not registered 76 approved 48 did no approve and 28 had no opinion Represent these data in a 3 x 4 matrix No Opinion MY NOTES Republicans Democrats Independents Not Registered ASK YOUR TEAC

Use the following matrix to perform the operation if possible If not possible enter IMPOSSIBLE into any cell of the matrix T0022 1535 2303 4 MY N Find 3B

Use the following matrices to perform the operation if possible If not possible enter IMPOSSIBLE into any cell of the matrix 48 H C D 51 Find C D MY M 13

8 Is it possible to diagonalize A diagonalize A 2 If yes diagonalize it and find A

6 Find the eigenvalues and the bases for the eigenspaces for the matrix A 1 2 21

4 Use matrix multiplication to find the image of the vector 3 2 when it is rotated clockwise about the origin through an angle of 60

5 Find the standard matrix for a single operator which performs the stated succession of operations compress by a factor of 1 3 in the y direction then shear by a factor of 2 in the x direction Is it possible to map the line y 2x 3 into another line using this standard matrix If yes find its equation

3 Suppose that x 1 x2 2 x3 4 x4 3 is a solution of a nonhomogeneous syster Ax b and that the solution set of the homogeneous system Ax 0 is given by the formulas X 3r 4s X2 T S X3 I X4 S a Find a vector form of the general solution of Ax 0 b Find a vector form of the general solution of Ax b

Find the missing side N 41 15 z

f x f x if if x x Type the left piece of the function Type the right piece of the function 3 1 color 33 X A 5

Rearrange solve for Vth How do I solve the equation to get the 2vth i know where the random numbers come from I just cannot remember how to solve the algebra part Uth2 3 C 3kT m 3k 4T 21 20th 2 500ms m 3kT m 1000ms

Find a such that the vector a a 1 belongs to Span 1 1 1 3 2 2 2 3 3 O O a 0 O a 2 O a 3 O a 1 a 1

Indicate whether the statement are True T or False F If A is non singular then A 0 If A is invertible then 4 1 A If A is invertible then A is invertible and 47 A Let A and B be matrices of sizes mxn and rxm respectively Then AB BAT Let A and B be matrices of sizes nxn then tr AB tr BA

5 20 Solve the following linear system of DE A 9x15x2 3x3 4x2 3x3 13 12 Iz

300 Let y 8 U The distance is 13 1 5 2 5 1 4 4 2 Find the distance from y to the plane in R spanned by u and 4 N N 2

se expansion by cofactors to find the determinant of the matrix W Z 24 18 10 32 18 24 32 22 35 40 y 30 15

1 7 1 17 5 Let A 1 118 8 18 0 9 8 18 0 9 B 1 18 be given 1 7 1 17 and Find an elementary matrix E such that B EA

The following graph shows a stock s actual market price and intrinsic value over time The intrinsic value comes from another researc analyst Use the dropdown menus to label the periods in which the stock was undervalued or overvalued Stock Price and Intrinsic Value 30 29 28 65 25 22 21 20 2013 B Overvalued 2014 2015 Years O 2016 A 2017 Actual Stock Price Intrinsic Value 2018

3 6 Let a matrix B is B 21 Find B6 by using a nonsingular matrix P such that D P BP is diagona

Supply p 9 8 Demand p 65 12q 2q a Sketch the first quadrant portions of those functions on the same set of axes Label the market equilibrium point E 80 60 40 20 80 60 P 40 P Supply Demand 1 Demand 2 3 E 4 DO 80 60 40 20 P 80 60 40 Supply Demand 1 Demand 2 3

Solve the equation logs 3x 1 16 18 x

Dints Given functions f x and g x find the composition functions x 4 a fog x and its domain b gof x and its domain

b through 4 3 and perpendicular to 3x 4y 24 i Standard form

Write an equation i in standard form and ii in slope intercept form for each line described below ac Show work leading to the answer to receive full credit a through 2 3 and parallel to 2x 3y 5 i Standard form

Use the distance formula and Pythagorean Theorem to determine whether the three points S T P are the vertices of a right triangle S 7 4 T 6 2 P 0 15

Consider the following equation f x x 2x 2 a Find the vertex of the graph of the equation x y b Determine whether the vertex is a maximum or minimum point maximum minimum c Determine what value of x gives the optimal value of the function X d Determine the optimal maximum or minimum value of the function

W Solve the equation by using the quadratic formula Enter your answers as a comma separated list If there is no real solution enter NO REAL SOLUTI 7w w 1 0 a Give real answers exactly b Give real answers rounded to two decimal places MY NOTES W ASK YOU

Find the vertex f x 5 x 1 12 Enter

f x 5x 5x 1 f 4 Be sure to simplify your answer Enter

Q5 Use LU decomposition method to find the unknowns in the given system of equations 2x 3y z 5 3x 2y z 10 habtx 5y 3z 0

Find the general solution of the equation y 2y 4y 8x 2 3x 1 2sinx 4cosx 6e 2x using the method of uncertain coefficients

The shaded region below is enclosed by the graphs of f x ex and g x x e 0 1 Region Between Two Curves Find the area of the shaded region A B 1 96 3 2 1 0 1 2

A matrix is given 54 Evaluate the eigenvalues and eigenvectors for given matrix 1940 a Eigenvalues 1 and 3 1 1 Eigenvectors b Eigenvalues 1 and 3 1 Eigenvectors c Eigenvalues 1 and 3 Eigenvectors d Eigenvalues 1 and 3 1 5 Eigenvectors Eigenvalues 1 and 3 1 1 Ficonvectore e

Consider the linear system X 2x 3x3 2x x 3x3 x x x3 10 0 01 A 0 10 0 L001 1 Write the linear system as augmented matrix Which one of the following is the reduced row echelon form of the system 5 1 3 100 2 B 0 10 0 001 11 100 2 C 0 10 0 1000 01

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