Matrices & Determinants Questions and Answers

Find the eigen values and eigen vectors of an operator given by the matrix 3 1 0 A 1 M 2 0 3 10 3 6 1 0 2 4 A A 3 9

Write the augmented matrix for the system of linear equations Do not perform any row opera 9x y 2 x y 9 11 10

of the matrix D are all integers d 3 5 d3 2 2 i ii D Xinx XXN XXX natrices 6 marks How to use this a i ii to create a pair of 3x3 matrices D D inverse where the entries are all integers

Use matrices to solve the system of equations if possible Use Gaussian elimination with back sub 3x 2y Z 15 x y 2z 17 x y 4z 27 x y z

24 Let V be the vector space of all 2 x 2 matrices and let S A1 A2 A3 A4 where 12 1 A 43 4 23 A2 1 A3 1 1 2 13 and 28 A4 0 As in Example 8 find a basis for Sp S

Write the system of linear equations represented by the augmented matrix Use variables x y z 1 1 3 2 1

The graph of the function f is shown to the right Match the function gix Choose the correct graph below DA odu 4 with one of the graphs below OC du 00

2 Find the inverses of the following matrices 3 marks each 13 M HU 1 2 1 2 M 3 M 43 1 2 4 8 M3 0 1 20 1

Choose the definition for the function x 1 x 2 x 2 a y x 2 b y x 1 x 2 x 2 x 2 c y d y x 1 x 2 x 1 lx 2 x 2 x 2 x 2 x 2 Enter

37 For the matrix 010 A 0 0 1 000 describe the images and kernels of the matrices A A and A geometrically 3

Encode or decode the given message as requested numbering the letters of the alphabet 1 through 26 in their usual order 24 Use the coding matrix A cryptogram 96 25 17 a BEAD b CARE c CURB d DARE 5 21 10 d and its inverse A 1 to decode the

find basis and dimension for given vector space v a b b c Mat x2 R a b c R

12 2 1 12 1 2 Recall A homogeneous system of equations is of the form Az 0 a State the number of variables in the homogeneous system of equations Number of Variables k b Find the reduced row echelon form of A and from this determine the number of free variables in the solution to the system of eq Number of Free Variables m c Consider the vectors EB DL and t 12 12 Looking at the results in part b the vectors are d Solve a system of equations to determine t and to such that 2 12 B HE 2 12 2

1 0 1 3 01 1 3 0 0 1 3 Numerically verify that each initial state vector xo has the given steady state vector 0 21 0 4 1 0 305 0 695 Let A Xo Axo x 0 41 0 57 0 41 0 6 Axo 0 38 0 02 Ax A x

Let A B O C 4 OB C 6 8 12 B Show the calculations that are used to find the entries for matrix AB Choose the correct answer below OA 12 2 9 10 and C 4 12 6 9 4 2 6 10 8 12 12 9 8 2 12 10 4 2 6 9 8 2 12 9 4 12 12 10 4 2 12 9 8 12 6 10 8 2 6 9 4 12 6 10 8 12 12 10 8 3 6 3 4 3 12 3 3 11 8 12 6 9 8 2 6 10 4 12 12 9 4 2 12 10 Show the calculations that are used to find the entries for matrix AC Choose the correct answer below OA 4 11 6 3 4 3 6 4 8 11 12 3 8 3 12 4 3 4 3 12 4 4 11 12 3 8 3 6 4 8 11 6 3 4 3 6 3 4 11 6 4 8 3 12 3 8 11 12 4 8 11 6 4 4 11 12 4 Verify that AB AC and yet B C Verify that AB AC by simplifying AB AC Type an integer or decimal for each matrix element

1 point Let A P 1 0 0 3 12 6 0 1 0 12 6 3 6 Find an orthogonal matrix P and a diagonal matrix D such that D PT AP 6 6 0 0 18 H D 0 0 0 9 0 0 0 9

Find a basis for the row space column space and null space of the following matrices a A b B 1203 1 1 36 021 183 1 0 1 3 2 1 21 2 4 in Z5

The table gives the amount in billions of dollars of Country A s trade in selected goods and services with Country B and Country C in a certain year Use the table to complete parts a through d below Click the icon to view the table of Country A s trade a Write a matrix for Country A s trade with Country B Type an integer or a decimal for each matrix element b Write a matrix for Country A s trade with Country C EXTE Type an integer or a decimal for each matrix element c Use the matrices from parts a and b to write a matrix showing Country A s total trade with Country B and Country C Type an integer or a decimal for each matrix element d Use the matrices from parts a and b to write a matrix showing the difference in Country A s trade with Country B and Country C Type an integer or a decimal for each matrix element Country A s trade Food and Live Animals Beverages and Tobacco Crude Minerals Mineral Fuels Chemicals Manufactured Goods Machinery Country B Country A Country A Exports Imports 24 8 21 5 2 1 0 9 7 2 14 9 16 1 33 8 103 7 26 6 37 7 93 2 38 7 133 8 Country C Country A Country Exports Impor 14 8 10 2 0 7 3 5 7 4 2 4 24 8 25 5 26 4 92 6 41 8 5 8 18 5 154 2

Given the matrices below and that A LU complete L 1 U 1 and A A L A 1 1 3 6 19 2 9 1 Ex 42 1 12 0 1 0 1 L 1 0 0 6 1 0 2 3 1 U 0 0 U 1 3 0 1 0 20 0 5 1 0 1

Given the matrices below and that A LU solve Ax b A L X 1 3 6 17 4 9 1 4 Ex 42 0 1 3 1 2 13 10 b 18 107 65 1 3 U 0 1 00 2 1 1

Use the elementary matrix E to find EA using an elementary row operation E EA 0 0 10 1 00 0 1 Ex 5 A 2 3 5 2 7 4 1 3 9

Find an explicit description of Nul A by listing vectors that span the null space 17 2 6 1 A 01 3 10 00 0 00 A spanning set for Nul A is

Given matrix A shown below find 2A A Give your answer as a matrix using the matrix tool 2 1 9 68 4 4 3 6

Let B be a symmetric 5 x 5 matrix such that det B 2 Find adj 2B BT

Use elementary row operations to obtain an upper triangular matrix 5 10 15 6 19 12 2 and use it to compute det A if A 3 2

For the matrix Part a 4 points Part b 3 points Part c 4 points 1 B 1 2 2 1 1 Find det 3B Show all your work Evaluate B 3 B Use Cramer s rule to solve the following system for y only X B y 4 3 0 0 2

For the matrix 3 P 2 1 0 2 2 Part a 4 points Find det 2P Show all your work Part b 3 points Evaluate PT 5 P Part c 4 points Use Cramer s rule to solve the following system for x only P

After having defined the place of intersection by coordinates by an equation by a terminal of the following two groups of plans explain the main difference between these places of intersection Groupe A 2x y 3z 6 T2 x 2y 3z 1 3 x 5y 12z 3 Groupe B T4 3x 2y 2z 1 5 2x y z 3 6 2x 6y 2z 2

Decide if the following statements are TRUE or FALSE Write a proof for the true nes and provide a counter example for the rest Every linear operator T R Rn can be written as T D N where D is diagonalizable N is nilpotent and DN ND

2 9 Given that cos a 4 a sin sin a b 15 and both angles terminate in quadrant IV find

Consider the following matrix 2 1 4 A 1 1 1 104 Choose the correct description of A Find A if it exists A is nonsingular That is it has an inverse OA is singular That is its inverse doesn t exist 00 000 888

c Find the product AB where A and B are matrices given by 3 B 1 4 15 21 1 3 51 12 4 3 4 6 A 12

5 x 5 5 Consider the one to one function f x a Find the inverse of the function b Graph both the function and its inverse on the same coordinate plane 101 8 6 10 8 6 4 2 A 2 2 14 6 2 6 8 10

Use the 5 4 1 of the function y f x below to answer the questions 5 3 y H a Is f 2 positive Yes No b For which value s of x is f x 0 Write your answer using interval notation 0 c For which value s of x is f x 0 If there is more than one value separate them with comman

Total personal income of the country in billions of dollars for selected years from 1960 to 2005 is given in the table a These data can be modeled by an exponential function Write the equation of this function with x as the number of years after 1960 b If this model is accurate what will be the country s total personal income in 2010 c In what year does the model predict the total personal income will reach 21 trillion Year 1960 1970 1980 1990 2000 2005 Personal Income 413 6 833 4 2310 4 4875 6 8424 7 10 230 6

Problem 6 10 points Let G 0 Find the minimal possible value of Ax b for x E R

10 9 8 7 6 5 4 3 2 1 10 9 8 7 6 5 1 2 3 6 7 9 10 1 3 4 5 6 7 8 9 10 1 Determine if each of the following points is a solution to the system of inequalities shown above Explain your reasoning A 6 1 B 2 5 C 0 0 2 Plot two points that are included in the solution set of the system of inequalities Label the points and B

Use the matrices below to perform matrix multiplication A 3 5 1 8 2 6 AB 56 B 1 3 2 4 4 1

9 Use the coding matrix A A CARE B CURB C BEAD D DARE 2 1 25 13 and its inverse A 3 1 5 2 to decode the cryptogram 9 6 25 17

the best answer for the question 19 Use the coding matrix A A ALAS B ARMS C ABLE D ACTS 12 10 and its inverse A 92 to decode the cryptogram 7 8 16 211

A graph G is a k regular graph if all the vertices of G has the same degree k For example Kn is a n 1 regular graph Part A Let G X Y E be a regular bipartite graph prove that X Y Part B Use Hall s theorem to prove that if G X Y E is a regular bipartite graph then there is a matching of size X Part C Let G X Y E be a k regular bipartite graph then the edge set of G can be partitioned into k matchinga which do not share any common edge Hint you may want to use induction

Given a training data set comprising N observations x where n 1 N together with corresponding target values t i 15 points Consider the following sum of squares error function N Ep w tn w o x n 1 Find an expression for the solution w that minimizes this error function 1

Find a basis for the subspace of R4 spanned by S S 5 9 5 53 2 5 2 5 8 2 8 17 0 2 0 15 300

Use the matrices below to perform matrix multiplication 5 6 A 2 2 11 BA B 1 2 7 2 2 1 6 3

Use the matrices below to perform the indicated operations 4 3 4 1 1 B 4 4 A AB 3 7 2

Find a logarithmic model for the data set algebraically Round all numbers to three decimal places y x Expenditures billions X 1872 6 2397 6 2886 5 3061 6 Year y 1990 1995 2000 2001 Expenditures billions X 3240 8 3424 7 3620 6 3877 2 Year y 2002 2003 2004 2005

Use the matrix below to perform scalar multiplication B 4B 4 2 58 8 6

Use the matrix below to perform the indicated operations A 2 A A 100 0 0 1 010 Get Hel E

9 14 7 12 each of the following vectors if the vector is an eigenvector for A determine the corresponding eigenvalue If the vector is not an eigenvector for A enter NONE Dose A The vector The vector The vector The vector The vector 5 8 4 8 4 2 corresponds to the eigenvalue corresponds to the eigenvalue corresponds to the eigenvalue corresponds to the eigenvalue corresponds to the eigenvalue

Find the state vector x3 for the given stochastic matrix A and initial state vector xo 9 16 13 31 4 15 13 31 1 3 3 19 16 19 XO