Matrices & Determinants Questions and Answers

The equation of the piecewise defined function f x is below What is the value of f 1 x 1 45x 1 f x x 1 x 2 3x x22 Of 1 2 Of 1 1 Of 1 2 Of 1 3

The hourly wages earned by 20 employees are shown in the first box and whisker plot below The person earning 1 per hour quits and is replaced with a person earning 8 per hour The graph of the resulting salaries is shown in plot 2 7 8 9 10 11 12 13 14 15 Plot 1 7 8 9 10 11 12 13 14 15 Plot 2 How does the mean and median change from plot 1 to plot 2 The mean and median remain the same The mean decreases and the median remains the same

11 12 Evaluate the given f w t 2

Based on the family the graph below belongs to which equation could represent the graph O G y x 2 3

If f x x 7 and g x XIX 6 O XIX 6 XIX 13 XIX 13 o 1 x 13 what is the domain of fog x

Determines whether the images below are linear images You must give ground answer If an answer is yes then write the standard matrix for the linear representation T R R 2 x1 1 x1 2x X1 x 7x

Suppose A is a square matrix such that det A 2 How many solutions does Ax 0 1 2 0 hav

Show the determinant of the following matrix 0 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0

linearly independent ones Type 0 for linearly dependent type 1 for linearly independent For some of these sets of vectors you can determine whether or not they are linearly independent without performing row reduction H 2 2130 00 39 41 10 16 08 BOA 1 13 BOE

4 If a square matrix A of order n satisfies A 0 where k is a positive integer prove 1 7 points A is not invertible 2 3 points I A is invertible and its inverse is I A I A A Ak 1

6 For each of the following matrices A describe the associated matrix transformation TA geometrically Determine what it does to the standard basis vectors of R and draw a picture Then describe what it is doing in your own words in a sentence or two a A b A c A 0 0 1 100 01 0 1 0 01 0 0 10 000 8

Let f x x 5 a Find f x b State the domain and range for f x

olve for A ou can resize a matrix when appropriate by clicking and dragging the bottom right corner of the matrix 63 2 1 7 6 3 38 1 To 201 3 69 3 1 1 1 0 6

en the following matrices A and B find elementary matrices E and F such that B FEA u can resize a matrix when appropriate by clicking and dragging the bottom right corner of the matrix 8 7 7 9 20 5 6 5 4 3 4 B 8 7 7 5 6 5 49 37 35 4 3 4

Consider the following system of linear equations x1 x2 2x3 x4 0 x1 2x2x3 1 x1 x2 8x3 a 2 2 x4 2a where a is a real number For what values of a does the system have a no solution b a unique solution and c infinitely many solution Clearly justify your

2 10 points Show that the following limit does not exist Hint Go to 1 1 along the path x 1 y x and compare y x lim y 1 y ln x

Worksheet Triangle Sum and Exterior angle Theorem I Find the value of x 1 x 47 58 X 2 X 96 21 Name 3 X Period 3x 1 31

7 4 1 2 equation Ax x Let matrix A and vector x 4 find the constant value that satisfy the

Find the general solution of the simple homogeneous system below which consists of a single linear equation Give your answer as linear combination of vectors Let x and x3 be free variables 2x 6x 8x3 0 O A O C X1 X X 4 X2 EHEHE X2 3 X X3 X3 X3 with X2 X3 free X X2 X 0 X3 with X2 X3 free X3 O B O D X BHOHE X2 X2 X3 with X2 X3 free 5 X2 BHOHO X x3 with X2 X3 free 4 0

1 Consider the 6 x 6 matrix A 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 a 2 marks Compute A Hint Partition A as a block matrix to save yourself a lot work b 1 mark What do you think A2023 equals You do not have to rigorously prove yo in corrent but hrinflu oulain who think it s t

Order Naproxen Sodium 0 75 grams Available Naproxen Sodium 375 mg per tablet How many tablet s will the nurse administer

b B Let G A E GL R A d for some a b E R Prove that G a forms a subgroup of GL2 R the group of 2 x 2 invertible matrices with real entries Can you identify this group as something familiar Let be a closed binary operation on a set S and suppose that is associative and there is an identity element e In this case e call S under a semigroup Thus a group is just a semigroup where each element has an inverse

Let AX B be a system of 7 linear equations in 6 variables and C its augmented matrix If the rank of A is 4 and the rank of C is 5 then which of the following statements is true Select one Select one There are no such system The system has no solution The system has a unique solution The system hon infinit inconsistent custom

1 a Find the matrix P that multiplies every vector r y 2 R to produce the vector 3x 2y z 5y 2 8r Also find P 1 b Find the matrix P that multiplies every vector r y R2 to produce 5x 4y 2x 3y 2x R

Select the simplified form of the difference 6n 8n 3 3n 2 8n 2 3n 4 6n 3

Question 7 What is the result of the following matrix 2 5 6 2 5 3 2 3 1 3 1 5 1 2

Express the following invertible matrix A as a product of elementary matrices You can resize a matrix when appropriate by clicking and dragging the bottom right corner of the matrix A 01 20 Number of Matrices 1 000

e 3 Marks With the given enciphering matrix 13 Find the ciphertext for

9 8 points Considering the following matrix A which are the singular values of A a 1 b 2 c 3 d 4 e 9 0 1 0 0 0020 0003 0000

Given 2 x 3 matrix M find a 3 x 2 matrix B such that MB I where I is the 2 x 2 identity matrix 1 00 2 00 3 00 2 00 2 00 4 00 B 1 2 2 2 M X 0

2 6 The vectors 1 5 0 1 21 2 02 1 2 1 4 relative to the 1 6 lattice basis Determine the coordinates of this site x 1 2 x3 relative to the standard basis of R 3 03 the crystal lattice of titanium One of the octahedral sites is Select Select 0 3 0 Select 2 0 A 0 in R form a basis for the unit cell of 4 8

a 325 cm 622 cm 409 cm Triangular Right Prism with Right Triangular Bases b 15 ft 45 ft 22 ft 12 ft Trapezoidal Right Prism

13 2 3 Quiz Dot Plots Box Plots and Histograms The frequency table shows the results of a survey asking people how many hours they spend online per week On a piece of paper draw a histogram to represent the data Then determine which answer choice matches the histogram you drew 2109 11 Hours online Frequency 5 8 0 3 4 7 8 11 12 15 16 19 Hours online 10 9 7

7 1 11 Completely factor the following expressions if possible HINT LOOK FOR GCF FIRST 9 4k2 14k 12 11 2x y 8x y 6xy

Let A 3 1 3 B 0 7 2 C 9 3 3 and D be four points in R such that AC and BD are orthogonal and AD and BC are parallel Using vector method a find the coordinates of D b c find two unit vectors which are orthogonal to the plane containing A B and C find the area of the triangle ABC

Find the lengths of the missing sides if side a is opposite angle A side b is opposite angle B and side c is the hypotenuse c 14 4A 45

Consider a linear system whose augmented matrix is 1 3 0 2 4 3 12 k 20 1 1 5 For what value of k will the system have no solutions k

5 points Express the matrix A 4 1 4 6 19 as a product of elementary matrices

13 0 77 points References Exercise 2 2A Algo Effect of accrued LO 2 2 2 3 During Year 1 Chung Corporation earned 6 800 Required Based on this information alone Hint Record the the requirements a Prepare the December 31 Year 1 balance shee b Determine the amount of net Income that Chun

Two woodworkers Tom and Carlos earn a profit of 86 for making a table and 65 for making a chair Tom must work 3 hours and Carlos 2 hours to make a chair Tom must work 2 hours and Carlos 6 hours to make a table If neither wants to work more than 42 hours per week how many tables and chairs should they make to maximize their profit

3 Consider the matrix A 1 0 3 3 2 1 5 6 Suppose B is the reduced row echelon form RREF of A Find a matrix U such that B UA

Solve the following linear system using elementary operations 5x 9y X Ty Answer x y 85 27 26 5 X

Suppose a 2 2 matrix A satisfies and for some a ER 44 4 A a 40 4 A a a Determine an invertible matrix P which diagonalises A and the corresponding diagonal matrix D b Use the diagonalisation from a to calculate A5 Of course this can be done in other ways so your answer must clearly show how you have used the diagonalisation you should not need to find an expression for A

etermine the following entries 12 34 A 30 0 3 4 1 3 1 32 24 2 4 2

Consider the accompanying matrix as the augmented matrix of a linear system State in words the next two elementary row operations that should be performed in the process of solving the system O A Replace row 2 by its sum with times row 4 Type an integer or a simplified fraction Interchange row 3 and row 2 OB C Scale row 1 by Type an integer or a simplified fraction OD Replace row 4 by its sum with times row 3 1 0 0 0 What should be the first elementary row operation performed Select the correct choice below and if necessary fill in the answer box to complete your choice Type an integer or a simplified fraction 5 3 0 0 5 7 1 4 14 2 0 2 0 5 4 2

3 3 2 points If possible write the vector 5 as a linear combination of the vectors AND 4 2 points If possible write the vector a as a linear combination of the vectors 3 and

15 Which of the following a b c V2 eigenvector 6 N 0 1 2 2

2 Lay 1 1 17 For what value s of h is b in the plane spanned by a and a2 D A 0 3 7 2 b h

A be a nonsingular matrix If I A and I A are also nonsingular matrice w that I A I I A

a Solve the following linear system 1 1 HOBBHO 0 1 2 0 01 20 0 1 1 0 32 1 12 13 b If the system in a above is Ax b where the left hand side has the LU decomposition of A i Write the system in a in the form Ax b ii Perform two iterations of the Successive Over Relaxation method with 0 3 to approximate the solution of Ax b starting with 20 21 22 0 0 0