Matrices & Determinants Questions and Answers

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

Use mathematical induction on integer n to prove each of the following statements 7 c 1 3 4 1 2n 23n n 12 5 2n 1 1 4n for m 1

Problem 3 2 points Find a basis for Span 1 2x 2x x 1 x 1 x in V

Find the unique n 1 th degree polynomials that pass through the given sets of n data points The data points 1 0 0 1 1 2 2 3

point Find two basic solutions to the following system of homogeneous equations x y 2z 4w 0 4x 2y z w 0

Which is a solution for 4x 8 3x 14 A 2 22 OB 2 7 C 3 D 6 O E 7

10 marks 1 Given the linear system x 2y 3z 1 2x By z 1 3x y 2z Find the values of a and B if any such that the system a has one single solution and give the solution values of x y z b has infinite solutions and give the solution values of x y z c has no solutions Verify your answers

Find all zeros of f x 8x 4 20x 3 10x 2 5x 3 stating multiplicity when appropriate

b c a b c 1 2 0 12 1 3 1 7 122 3 3 4 5 L

Let D 1 2 N and D 2 YN two sets data such that 3x 29 11 Yi 1 If MAD 12 find MAD2 2 If If MAD 12 find MAD2 3 If o 22 find o 4 If If o2 16 find o for all i 1 N 1

c What do you understand by the term LU decomposition Write the matrix below in the form A LU 1 5 3 2 1 1 1 where A 0 4 7 Marks

Solve the following inequality Write the solution set using interval notation Express all values as integers or reduced fractions Enter DNE if no solution exists 1 5 5z 2 3

Find the equation in terms of x of the line through the points 1 7 and 2 2 Y Question Hol

36 6x 3x 2

Graph the line passing through 5 3 with slope m 4 14 12 10 8 6 4 2 14 12 10 8 64 2 24 8 10 12 14 246 8 10 12 14

Find the reduced row echelon form of this augmented matrix 1 2 0 150 1 1 3 200 0 0 1 250

Write the system of equations as an augmented matrix Then use the Gauss Jordan method to solve for x and y x 5y 3 2x 3y 1 Solution Enter the solution as an ordered nair y y with no r

The given matrix is an augmented matrix representing a system of linear equations Use the Gauss Jordon elimination method to find the solution of the system 7 6 5 01 4 06 4 100 01 0 001 56 1 26

Write the system of equations as an augmented matrix m c 100 250 a 150 C

Convert to Scalar Form Determine how and intersect x 1 4s T y 5 6s 6t and 3x 2y 4z 17 0 1 z 1 3t

By locating the pivots find a basis for the column space of 0 5 4 3 0021 0000 00 U

8 Given a b 0 show that the matrix B a 0 88 b has operator norm a Compute its Hilbert Schmidt norm

7 Identify the singularities of 2 1 z 22 x 2 z and find the Laurent series for this function for each distinct annular region In each region gi general formulas for the series coefficients an bn

11 Admission tickets to the Braden River Little League all star game cost 4 00 for adults and 1 50 for children A total of 225 tickets were sold and a total of 500 was collected from ticket sale How many adult tickets and how many children s tickets were sold Set up a system of equations an solve

3 Solve the systems 3x 4y 2z 15 2x 4y z 16 2x 3y 5z 20

1 Solve the systems x y z 14 2y 3z 14 16y 24z 112

1 point Let T be an linear transformation from R to Rs Let A be the matrix associated to T fill in the correct answer for each of the following situations 1 Two columns in the row echelon form of A are not pivot columns 2 The row echelon form of A has no column corresponding to a free variable 3 The row echelon form of A has a column corresponding to a free variable 4 Every column in the row echelon form of A is a pivot column A T is not one to one B T is one to one

write an equation for the graph The equation for the graph is O A NONA O 6 4 2 2 4 6 2

2 Which of the following sets of vectors are independent i 2 2 3 2 2 4 4 4 6 ii 2 2 3 2 2 4 6 6 11 iii 2 2 3 2 2 4 10 10 17 A ii only B iii only C i only D i and iii only E i and ii only F ii and iii G all of them H none of them

For the function g x 42 3 evaluate and simplify a g 9 4g 3 b g x 8 4x 29 c g x g 8 0 o

E 11 Given a vector A 3a 2a 16a find its components normal to the plane 2x 2yz 8 0 and tangential to it

Find the focus directrix vertex and axis of symmetry for the parabola 12 y 2 z 1 Focus 1 1 Directrix y 5 Vertex 1 2 Be sure to enter each answer in the appropriate format Hint What is the appropriate notation for a line or a point Graph the parabola Include the directrix and focus with your graph 63 43 2 L r 9 54 4 3 23 1 7737 2 2 A 45 6

The island arc running from the Virgin Islands to Trinidad and Tobago is called O The guianas O The Lesser Antilles The Bahamas O The Greater Antilles

1 Given and 1 3 1 0 find AB and BA 3 AB BA 2 Hint Matrices need to be entered as elements of row 1 separated by commas elements of row 2 separated by commas elements of each row separated by commas Example C 1 2 3 3 would be entered as 1 2 3 4 5 6 4 5 6

Write the system of equations as an augmented matrix 2a 4c 4r 68 6a c 5T 48 7c 14 a 7r

Write the system of equations as an augmented matrix 24 3c 300 100 350 2u y 9c C

Given the matrices A A It is Select an answer Select an answer B If ad possible not possible 5 3 4 3 2 1 3 4 5 to add these matrices at is A B and B 2 5 4 5 4 2 1 5 4

Let A and B 10 2 6 8 5 6 1 3 9 10 then use the matrices above to perform matrix multiplication AB

4 2 45 2 1 2 T 2 3 33 5 T 11

Matrix A has the dimensions 3 x 6 and Matrix B has the dimensions 6 x 4 What are the dimensions of the product AB X note if multiplication is not possible put 1 in for both the row column dimensions

Find the roots of the following quadratic equa tions by factoring 7 x 2x 1 0 8 x 8x 15 0 9 x 1 0 10 x 6x 8 0 11 x 4x 21 0 12 x 4x 12 0 13 x 10x 24 0 14 x2 11x 24 0 15 x 4x 5 0 16 x 10x 24 0

Decide if the following statements are true 1 y 2 is a root of y y 6 0 2 x 1 and x 3 are roots of x 4x 3 0 3 x 2 and x 6 are roots of x 4x 12 0 4 y 1 and y 3 4 are roots of 4y 7y 3 0 5 x 1 and x 2 3 are roots of 3x 5x 2 0

Given the system of equations 2x 3y 9z 15 1 3z 3 3x y 4z 2 a Determine the type of system inconsistent dependent o b If your answer is dependent in a find the complete solution Write x y as functions of t where z t If your answer is inconsistent write DNE for all three variables 49 I y 11

1 3 marks Find a formula for the nth term of the sequence defined by the third order recursion tn 3 7tn 1 6tn with initial values to 2 t 8 t 0 Calculate t3 directly and use it to check your formula n 0 1 2 3 tn 2 8 0

Ricardo invests 4100 in two different accounts The first account paid 8 the second account paid 5 in interest At the end of the first year he had earned 292 in interest How much was in each account s at 8 at 5

The shape of this graph is best described as symmetric skewed constant peaks and valleys 12 10 8 6 4 2 CO 666 0 100 199 2012 Housing Prices 200 299 300 399 400 499 500 599 600 699 IL 700 799 668 008 Sales Price Thousands of Dollars

5 Let a b E R and define a matrix by a 1 T E b a Is T necessarily invertible Explain your reasoning b Show that for all a b R T has two distinct real eigenvalues c Now suppose a b Find an eigenvector for each eigenvalue of 7

Determine the scalar equation i e standard form of the plane with vector equation 16 0 0 0 5 s 2 t 3 2 5

Pivot the matrix about the circled element 26 0 4 3 0 2 13 The result of pivoting is Type an integer or simplified fraction for each matrix elemen on for each Pivot the matrix about the circled element 2542 1 58 4024 The result of pivoting is Type an integer or simplified fraction for each matrix element

7 Use the Remainder Theorem to evaluate f x 6x4 x3 15x 2x 7 at x 2