Matrices & Determinants Questions and Answers
Algebra
Matrices & DeterminantsPart D Application 20 marks Full Solutions are required Consider the vectors u and w If 2 4 8 4 2 1 1 5 and W 1 1 1 find xw 3 marks 8
Algebra
Matrices & DeterminantsIf possible find AB If not possible enter IMPOSSIBLE in any cell of the matrix 0 12 483 8 05 B 7 1 8 AB A 000 11 State the dimension of the result If not possible enter IMPOSSIBLE in both answer blanks
Algebra
Matrices & Determinants7 Suppose A is an n x n invertible matrix and A is an eigenvalue of A Suppose x is an eigenvector of corresponding to A a Show that A0 and x is an eigenvector of A corresponding to eigenvalue A b Define a 2n x 2n matrix C and the vector y in R2 by A A 1 c A4 Is y an eigenvector of C If yes what is its corresponding eigenvalue y
Algebra
Matrices & DeterminantsLet A be a diagonalizable n x n matrix and let P be an invertible n x n matrix such that B P AP is the diagonal form of A Prove that Ak PBkp 1 where k is a positive integer Use the result above to find the indicated power of A 4 0 4 A4 A 5 6 1 4 A4 0 6 000 00
Algebra
Matrices & DeterminantsWrite the system of linear equations as a matrix equation AX B and use Gauss Jordan elimination on the augmented matrix 4 B to solve for the matrix X X1 x 3x3 6 8x 7x 8x3 104 6x1 6x 8x3 14 a C b 1 1 3 8 78 d e 797 1 1 3 6 8 7 8 104 6 6 8 14 6 8 8 EH LH 6 L6 6 8 5 1 3 1 1 3 8 7 8 L6 6 8 L 14 con 8 8 7 8 6 6 8 5 1 1 3 8 7 8 6 6 8 14 6 104 14 0 5 00 6 104 104 8 5
Algebra
Matrices & DeterminantsLet the linear transformation T R R3 and S R3 R4 be defined by T x y x 2x y 3x 4y S x y z 2x z 3x z x y z If possible then find SoT
Algebra
Matrices & DeterminantsLet A 3 4 1 2 5 3 and b Define a transformation T by T x Ax If possible find a vector x whose image under T is b Otherwise state that b is not in the range of the transformation T 10 5 5 18 0 1 3 27 9 0 3 3
Algebra
Matrices & DeterminantsProblem 7 7 points et W x y z R 3x 2y z 0 Determine an orthonormal basis of W G 3 14 2 14 1 14 3 14 2 14 1 14 A B D E 9 3 14 2 14 1 14 0 0 2 13 13 2 2 10 0 2 10 6 2 10
Algebra
Matrices & DeterminantsLet y and u B Write y as the sum of a vector in Span u and a vector orthogonal to u y tr Type an integer or simplified fraction for each matrix olement List the terms in the same order as they appear in the original list
Algebra
Matrices & Determinantsof the matrix Find the characteristic polynomial using either a cofactor expansion or the special formula for 3x3 determinants Note Finding the characteristic polynomial of a 3x3 matrix is not easy to do with just row operations because the variable y is involved Find the characteristic polynomial of the matrix using eithe involved 043 401 310 The characteristic polynomial is Type an expression using as the variable
Algebra
Matrices & Determinants1 point Calculate the area of the parallelogram determined by the following vectors Area 5 2 0 4
Algebra
Matrices & DeterminantsLet A be the following matrix A 0 1 1 0 a Let A R x3 What is the rank of A b Let F be the finite field of two elements and let A F3 What is the rank of A2 15
Algebra
Matrices & Determinantsii Consider the equation T and hence 1 1 2 5 I 1 3 2 2 4 3 Using our results in 1 we see that the determinant in the left hand side of the equation is equal to 6 3 2 0
Algebra
Matrices & DeterminantsProblem 7 8 points Consider the ordered bases B a Find the transition matrix from C to B 188 b Find the coordinates of u 4 7 and 1 6 C V B 2 in the ordered basis B 18 c Find the coordinates of v in the ordered basis B it the coordinate vector of vin C is vic 2 for the vector space R Note You can earn partial credit on this problem
Algebra
Matrices & DeterminantsFind the matrix P that orthogonally diagonalizes the matrix A A B D P P P P 3 3 2 6 3 2 3 3 3 3 3 2 3 3 2 6 0 6 6 2 6 w w wo ala ala wadada 2 2 3 3 2 2 0 3 3 3 6 6 6 3 0 3 3 3 3 3 3 2 2 0 6 6 6 3 5 11 1 5 1 1 15
Algebra
Matrices & Determinants1 Let A a xn be symmetric and positive definite Show that a a 0 i 1 2 n b a a a i j 1 2 n
Algebra
Matrices & Determinantsc Let B 3 3 41 04 i Use Givens rotations to reduce B to a row echelon form Your answer should include each Givens matrix you use ii Find the QR factors of B where is an orthonormal matrix and R row echelon
Algebra
Matrices & DeterminantsUse technology to solve the system of equations if a solution exists If there are infinitely many solutions express your answers in terms of x4 as in Example 3 there is no solution enter NO SOLUTION X 3x 2x3 2x4 3x3 x x 2X1 2x3 X X1 X2 X3 X 4 3x2 3x4 3 1 ASK YOUR TEAC 1 1
Algebra
Matrices & DeterminantsUse the following matrix to perform the indicated operation when possible If not possible enter IMPOSSIBLE into any cell of the matrix 0 1 3 532 023 A Find A
Algebra
Matrices & DeterminantsUse the matrices below Perform the indicated operation 8 5 3 1 0 0 5 8 6 010 1 6 7 0 0 1 A Find AI
Algebra
Matrices & Determinants3 Are U and V orthogonal with respect to standard inner product on M22 If not find the cosine of the angle between them Find d u v v 23 U 1 1 3 V
Algebra
Matrices & Determinants2 Using the inner product on R2 generated by matrix A 3 find u v for the vectors u 1 2 and v 2 4
Algebra
Matrices & DeterminantsThe three vectors 2 3 and n3 define the three planes 3 x R n x 6 x R n x 6 x R n3 x 10 What is the dimension d of their intersection 4 5 ii Explain and iustify the steps that you took to find this answer d
Algebra
Matrices & Determinantsfor the matrix A below find a value of k so that A has two basic eigenvectors associated with the eigenvalue A 3 k 0 0 0 4 14 1 4 12 0 3 6 0 0 1
Algebra
Matrices & Determinants1 Choose three different non zero integer values for the empty boxes and find the determinant of the resulting matrix 4 marks A 2010 420 3 1 5 3 0 2 4 4 0 2 4
Algebra
Matrices & Determinants3 Use SOHCAHTOA to find the missing side a and angle b 5 marks 3 b Height of kite 10m Length of string 30m a Find b AC Anglex 4 Use the Sine law to find the missing side in part a and angle in part b 5 marks
Algebra
Matrices & DeterminantsFor the matrix of transition probabilities 0 6 0 1 0 11 P 0 3 0 7 0 1 0 1 0 2 0 8 find p2x and p x for the state matrix p x p x 100 X 100 800 X 00 Find the steady state matrix X for P
Algebra
Matrices & Determinantsand a bass p 2 g for the vector space f x P 1 f 1 where Paarl is the vector space of polynomials in a with degree less than
Algebra
Matrices & Determinantsc Show that the basis of R is orthonormal 2 2 0 U W 2 2 2 6 6 3 V 2 21 3 3 Find the coordinates of 1 2 3 with respect to this basis
Algebra
Matrices & Determinantsnd a basis for the row space and the column space of the matrix 1 3 2 5 7 2 36 k of each basis Write each basis in reduced echelon form What is the 4 marks
Algebra
Matrices & Determinantsfind an invertible martix X and a diagonal matrix D such that XAX D Let 3 0 25 1 Find an invertible matrix X and a diagonal matrix D such that X AX D A X 1 88 D
Algebra
Matrices & DeterminantsMatrix A below gives the fraction of the earth s area and the projected fraction of its population for five continents in 2050 Matrix B gives the earth s are square miles and its projected 2050 population t A B Fraction Fraction of of Area Population 0 162 0 119 0 066 0 065 0 298 202 0 0471 North America 0 086 South America Europe 0 582 Asia Area 57 850 000 11 0 215 0 Africa Population 0 9 322 000 000 Population Find the area in square miles and population of each given continent by finding AB Area MY NOTES ASK YOUR
Algebra
Matrices & DeterminantsConsider the following Find CD 39 103 C 12 0 9 88 D E 0 Find CD E Find DE Find C DE Are CD E and C DE equal Yes CD E and C DE are equal No CD E and C DE are not equal
Algebra
Matrices & DeterminantsUse the following matrices to perform the indicated operation when possible If not possible enter IMPOSSIBLE into any cell of the matrix 0 3 1 B M A 543 IE 302 Find EAT X 11 49 7 27 1 102 PRACTICE ANOTHER 910
Algebra
Matrices & DeterminantsLet be the set of nxn real lower triangular matrices The dimension of as a vector space b n 1 2 c n n 1 d n e n 1 a n n 1 L tfen birini se in a e b c c a d b B d
Algebra
Matrices & DeterminantsLet F 0 1 2 1 6 7 6 a Are FFT and FTF defined Yes both FFT and FTF are defined FFT is defined and FTF is not defined FFT is not defined and FTF is defined No both FFT and FTF are not defined b What size is each product If the matrix is undefined enter DNE into both answer blanks FFT is and FTF is X c Can FFT FTF Explain Yes because FFT and FTF have the same dimensions No because FF is defined and FTF is not defined No because FFT and FTF have different dimensions No because FFT is not defined and FTF is defined X
Algebra
Matrices & DeterminantsUse row operations on an augmented matrix to solve the given system of linear equations X 2y 3z w 14 x 3y Z W 25 X y 8 2 X x y z w z w
Algebra
Matrices & DeterminantsA Use the following matrices to perform the indicated operation when possible If not possible enter IMPOSSIBLE into any cell of the matrix 0 3 17 543 302 Find EAT E MY NOTES 10 910 ASK
Algebra
Matrices & DeterminantsMY NOTES Use the following matrix to perform the indicated operation when possible If not possible enter IMPOSSIBLE into any cell of the matrix 79 c 3 C Find C3 AS
Algebra
Matrices & DeterminantsMY NOTES Find DE ASK YC Use the following matrices to perform the indicated operation when possible If not possible enter IMPOSSIBLE into any cell of the matrix 82 0 2 D 3 E 198 37
Algebra
Matrices & Determinantsnd a formula in terms of k for the entries of Ak where A is the diagonalizable matrix below and P AP D for the matrices P and D below 2 2 2 20 2 7 P 1 D 2 D 10 2 5 03 00
Algebra
Matrices & Determinants8 Assume that the matrix 001 A x 1 0 L1 0 0 has three linearly independent eigenvectors Then x A 1 B 0 D 2 C 1
Algebra
Matrices & Determinants3 a By solving a matrix equation express 8 as a linear combination of 0 and H 4 19 b Find the inverse of the matrix 1 2 4 634 A 1 2 2 3 mar
Algebra
Matrices & DeterminantsLet V 9 and B be the ordered basis B 1 H 5 Find the coordinates of the vector V relative to B
Algebra
Matrices & DeterminantsGiven the following matrices if possible determine 4A 3B If not state Not Possible 1 7 4 B 10 6 5 Given the following matrices if possible determine 4A 3B If not state Not Possible E B 6 9 5