Matrices & Determinants Questions and Answers

Algebra
Matrices & DeterminantsFind 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

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

Algebra
Matrices & Determinantsof 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

Algebra
Matrices & DeterminantsUse 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

Algebra
Matrices & Determinants24 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

Algebra
Matrices & DeterminantsWrite the system of linear equations represented by the augmented matrix Use variables x y z 1 1 3 2 1

Algebra
Matrices & DeterminantsThe 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

Algebra
Matrices & Determinants2 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

Algebra
Matrices & DeterminantsChoose 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

Algebra
Matrices & Determinants37 For the matrix 010 A 0 0 1 000 describe the images and kernels of the matrices A A and A geometrically 3

Algebra
Matrices & DeterminantsEncode 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

Algebra
Matrices & Determinantsfind basis and dimension for given vector space v a b b c Mat x2 R a b c R

Algebra
Matrices & Determinants12 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

Algebra
Matrices & Determinants1 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

Algebra
Matrices & DeterminantsLet 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

Algebra
Matrices & Determinants1 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

Algebra
Matrices & DeterminantsFind 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

Algebra
Matrices & DeterminantsThe 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

Algebra
Matrices & DeterminantsGiven 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

Algebra
Matrices & DeterminantsGiven 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

Algebra
Matrices & DeterminantsUse 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

Algebra
Matrices & DeterminantsFind 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

Algebra
Matrices & DeterminantsGiven 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


Algebra
Matrices & DeterminantsUse 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

Algebra
Matrices & DeterminantsFor 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

Algebra
Matrices & DeterminantsFor 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

Algebra
Matrices & DeterminantsAfter 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

Algebra
Matrices & DeterminantsDecide 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

Algebra
Matrices & Determinants2 9 Given that cos a 4 a sin sin a b 15 and both angles terminate in quadrant IV find

Algebra
Matrices & DeterminantsConsider 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

Algebra
Matrices & Determinantsc 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

Algebra
Matrices & Determinants5 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

Algebra
Matrices & DeterminantsUse 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

Algebra
Matrices & DeterminantsTotal 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

Algebra
Matrices & DeterminantsProblem 6 10 points Let G 0 Find the minimal possible value of Ax b for x E R

Algebra
Matrices & Determinants10 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

Algebra
Matrices & DeterminantsUse the matrices below to perform matrix multiplication A 3 5 1 8 2 6 AB 56 B 1 3 2 4 4 1

Algebra
Matrices & Determinants9 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

Algebra
Matrices & Determinantsthe 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

Algebra
Matrices & DeterminantsA 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

Algebra
Matrices & DeterminantsGiven 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

Algebra
Matrices & DeterminantsFind 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

Algebra
Matrices & DeterminantsUse the matrices below to perform matrix multiplication 5 6 A 2 2 11 BA B 1 2 7 2 2 1 6 3

Algebra
Matrices & DeterminantsUse the matrices below to perform the indicated operations 4 3 4 1 1 B 4 4 A AB 3 7 2

Algebra
Matrices & DeterminantsFind 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


Algebra
Matrices & DeterminantsUse the matrix below to perform the indicated operations A 2 A A 100 0 0 1 010 Get Hel E

Algebra
Matrices & Determinants9 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

Algebra
Matrices & DeterminantsFind 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