Matrices & Determinants Questions and Answers

T onsider the quadratic form q4 x y A where A I r the transformation T R R with standard matrix 10 04 x y ere 5 4 4 6 i e a vertical stretch by a factor of 4 find the transformed quadratic form qp x y obtained from via 9A

Question 3 12 points Solve each system State whether it is an inconsistent system or has infinitely many solutions If the system has infinitely many solutions write the solution set in terms of only the arbitrary variable z Show work a x y z 0 4x 2y6z 0 2x y 3z 0 b 2x y 4z 4 x 2y z 2 3x y 3z 1

4 2 points Let V R and let W W is a subspace of V or not answers 21 1 1 1 13 9 Decide whether the without a proper justification is not acceptable

The matrix A 3 14 21 46 21 42 14 42 38 is diagonalisable with eigenvalues 3 4 and 4 An eigenvector corresponding to the eigenvalue 3 is Find an invertible matrix M such that M 1AM Enter the Matrix M in the box below N 1 3 3 3 0 0 0 40 0 04 E 1 2 2 3 3 2

3 3 3 2 Write as a product of an orthogonal matrix and an upper triangular matrix Clearly show each step used in the process 4

Consider the following system of linear equations 2y 2z2w 2 7y 3z 4w q 3x x 4y p z pw 8 where p and q are real numbers Using Gaussian elimination determine all possible values of p and q such that the system a no solution b infinitely many solutions with one free variable c infinitely many solutions with two free variables Then solve the system in part c

Find the B matrix for the transformation xAx where B b b A 6 2 1 b 1 2 The B matrix of the given transformation is b

Define T P2 P by T p p 0 p 10 t p 11 t a Show that T is a linear transformation b Find T p when p t 11 t Is p an eigenvector of T c Find the matrix for T relative to the basis 1 tt for P a Let u t and v t be polynomials in P2 Complete the following explanation T u v Evaluate T u v

cos x 0 2n x Select 2 correct answer s 00 11x 6 5333 No solution 7 02 12 2 55

Ashley and Stephanie went shopping for Father s Day at the same store Stephanie spent 105 for 2 t shirts and 3 pairs of shorts Ashley bought 3 t shirts and 2 pairs of shorts for 95 What was the cost of a pair of shorts Hint The equation for Ashley s purchases is 3T 2S 95 First determine the equation for Stephanie s purchases and then solve the system of equations using matrices O 10 for shorts O 17 for shorts O 15 for shorts 25 for shorts

8 A 1 24 0 00 8 1 1 1 2 Find row reduced 10 2 Co 2 2 6 7 2 1 9 3 2 4 echelon form of the A B and C matrices

find the values of k for which the matrix B is invertible b 2 1 3 0 2 1 k 4 1

2 A fence has to be erected around the pool 2 feet away from the sidewalk The fence has two gates each 3 feet wide How long is the fence

Given the following matrices it possible determine AB Identify the dimensions of the resulting matrix and fill out the matrix if it exists If not state Not Possible Answer 2 Points A 4 3 1 B 5 2 1 2 x 3 Not Possible 3 Selecting the check box will replace the entered answer value s with the check box value Entering the dimensions of a matrix will display any boxes necessary to create the matrix Keypad Keyboard Shortcuts

Consider the following matrix 14 2 3 4 6 7 10 Step 3 of 3 Determine the value of c24 if possible If the element indicated is not in the matrix state None Answer 2 Points Let C Keyboard St Selecting a radio button will replace the entered answer value s with the radio button value If the radio button is not selected the entered ans used

Given the following matrices if possible determine A B C If not state Not Possible Answer 6 A 43 43 4 19 B 3 4 3 7 Selecting a checkbox will replace the entered answer value s with the checkbox value If the checkbox is not selected the entered answer is used Not Possible

Find the unit price of 70 lbs of honey for 123 99 Round your answer to the nearest cent if necessar Provide your answer below sund

he augmented matrix that corresponds to the following system of equations 2x 6y 8 2 3 4z3x y 5 2 x 2 2 4y

Question 6 Verify the matrix is symmetric Find the eigenvalues of the matrix For each eigenvalue find the dimension of the corresponding eigenspace 0 1 1 H 1 0 1 1 1 1

Determine the quadratic regression for the data set below Select the correct answer below 0 961 0 546x 3 01 O 0 961 0 546x 3 01 0 961x 0 546x 3 01 0 96120 546x 3 01 2 8 1 4 6 0 2 5 1 4 2 5 6 3 10

The change of coordinates matrix from basis B to basis C of R is given by P C B 3 The coordinates of v E R relative to the basis B are given by v B relative to the basis C v c Ex 5 7 8 4 1 7 9 3 8 6 4 5 Find v c the coordinates of v 8

the blank by performing the indicated elementary row operation s swer 4 6 8 81 8 6 4R2 R1 2

Consider the ordered bases B 1 361 636164 and C Criangular 2 x 2 matrices a Find the transition matrix from C to B TB 3 D Find the coordinates of M in the ordered basis if the coordinate vector of M in C is M c 2 B M B 2 C Find M for the vector space V of upper

The infinite series given by 1 1 3 r 2 is a divergent geometric series true or false

20 Let a a an b 5 b A 3 2a and B B t 3a b 26 36 Determine which of the following is false a Columns of AB are L D b ABr 0 has nontrivial solutions c Columns of BT AT are L D d AB x 0 has nontrivial solutions e none of these

In AABC we are told that a 13 4B 84 and 4C 57 Solve for b and c A b C 84 57 13 B

Thinking and Inquiry 4 How can we use transformations to manipulate the shape position and behavior of a polynomial function and how do these changes impact its graphical representation and algebraic properties

Find the Solution X 7 4 X 4 X 1 41 X 4

Solve Using elimination 4x 4y 12 8x 3y 2

Show that any two eligenvector of the symmetric matrix corresponding to distinct eigenvalues are orthogonal 0 1 1 0 1 0 1 0 9 Find the characteristic polynomial of A 21 A 2 72 2 71 18h 10 0 Find the eigenvalues of A Enter your answers from smallest to largest A2 A3 4 26 1 4 26 Find the general form for every eigenvector corresponding to 1 Uses as your parameter x Find the general form for every eigenvector corresponding to X Use t as your parameter x Find the general form for every eigenvector corresponding to 13 Use u as your parameter x3 Find X X X X Find X X3 X X3 Find x x X X3 Determine whether the eigenvectors corresponding to distinct eigenvalues are orthogonal Select all that apply Ox and x are orthogonal

Solve the problem below Show all your work You must solve each part completely and correctly to get the indicated points for that part Given the linear system 2x1x2 x3 2 x 2x x3 1 4x 3x2 3x3 4 3x x 2x3 3 a Using the method of Gaussian elimination find an echelon matrix for this system 2 points b Find the reduced echelon matrix 2 points c Solve the system using part a or part b 1 point

Find the coordinate matrix of x in Rn relative to the basis B B 1 1 2 1 1 1 4 3 1 2 0 3 1 2 2 0 x 15 8 6 X B

21 Find a basis for the subspace of R consisting of all vectors 2 such that 91 7x2 4x3 0 x3 Hint Notice that this single equation counts as a system of linear equations find and describe the solutions Answer 974

Solve 7 4 10 Enter an exact answer or round your answer to the nearest tenth Do not include in your answer Provide your answer below

The population of a town was 230000 For 12 years the population grew by 4 per year compounded continuously What was the population at the end of the 12 year period according to the exponential growth function Round your answer down to the nearest whole number and do not include units Providos

Two forces of 36 N newtons and 51 N act on an object at right angles Find the magnitude of the resultant and the angle that it makes with the smaller force The magnitude of the resultant is Newtons Round to the nearest integer as needed The angle the resultant makes with the smaller force is Round to the nearest integer as needed

Jse the inverse of the coeffici x 4y 3z 11 x 3y 2z 7 4x 5y 4z 8

em 2 Consider the symmetric matrix 9 0 0 0 3 0 M 3 0 0 4 Find constants c and c2 such that the quadratic form q 7 C q v C C2 My satisfies for all ER with the condition that there exists two vectors wo such that q u q w c w Enter the values of c and ca in that order into the answer box below separated with a comma

possible values for X X2 and x3 What is the minimum volume of traffic from C to A 7020728 843and A A 110372 1305 2023 07 15 2021 50 10 2023 07 15 202117b3728 7h372

Solve the system of linear equations by Doolittle s method 23 07 15 202117 3x 9x2 6x3 4 6 18x 48x2 39x3 27 2 Is the Doolittle s method being same as Cholesky s method if not what are the differences 9x127x2 42x3 9 0 25 2023 07 15 20218 same 203 5 marks

During an epidemic the number of people who have never had the disease and who are not immune they are susceptible decreases exponentially according to the following function where t is time in days f t 17 000e 0 04t Find the number of susceptible people at each time a at the beginning of the epidemic b after 9 days c after 4 weeks a The number of susceptible people at the beginning of the epidemic is Round to the nearest whole number as needed b The number of susceptible people after 9 days is Round to the nearest whole number as needed c The number of susceptible people after 4 weeks is Round to the nearest whole number as needed

5 Consider the network of one way roads depicted in the figure below where the numbers indicate the number of vehicles per hour 100 50 A 4 equations and 5 variables B 4 equations and 6 variables C 5 equations and 4 variables D 6 equations and 4 variables F None of the showO 200 150 Suppose that you were to set up a system of linear equations to describe this network How many equations would you have to write down and how many variables would be involved

1 pt With real numbers if a 1 then a n2 2 Verify this by finding a 2 x 2 matrix A 1 This does not generalize to n x n matrices with I but A2 I

A P and D are n x n matrices Check the true statements below A If A is invertible then A is diagonalizable B If A is diagonalizable then A has n distinct eigenvalues C A is diagonalizable if A hasn distinct eigenvectors D If AP PD with D diagonal then the nonzero columns of P must be eigenvecto

Let A be a 3 x 3 diagonalizable matrix whose eigenvalues are X 4 A 1 and X3 1 If V 1 0 0 V2 1 1 0 V3 0 1 1 are eigenvectors of A corresponding to 2 and 3 respectively then factor A into a product XDX with D diagonal and use this factorization to find

nd the distance between the point and the line given by the set of parametric equations Round your answer to three decimal places 7 8 9 x 2t y t 3 z 2t 2

Find 7 v by using the standard matrix and the matrix relative to B and B T R R T x y x y x y v 7 4 B 1 1 0 1 B 1 1 0 0 1 1 1 0 1 a the standard matrix T V 11 7 4 b the matrix relative to B and 8 in terms of the standard basis T v

14 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0 metre Below is a speed vs distance graph for a race car at the Interlagos circuit in Brazil If the curve represents f x which of the following would represent the curve for a car that started at the 200 meter mark and was 75 km h slower at every point a 50f x 200 b f x 75 200 c f x 200 75 Corr Speed km h 288 3 200 Senna S 400 Curva do Sol 600 800 Reta Oposta 1000 1200 Descida do Lago 1400 1600 Ferradura 1800 2000 Laranjinha Turn 8 Pinheirinho 2200 2400 2600 Bico de Pato Mergulho 2800 3000 d f x 200 75 Jun o 3200 3400 Subida dos Boxes 3600 3800 4000 4200

1 point Solve the equation B 32 8 t 4x 5y 92

Let S be the solid obtained by rotating the region shown in the figure about the y axis Assume a 3 and b 2 YA 3 0 Sketch a typical approximating shell y y ax x b 2 00 8 4 2 2 X 3 X 2 2 1 y 1 1 2 X