Matrices & Determinants Questions and Answers

4 The following cyclic subgroups are all isomorphic the multiplicative group i in C the additive group Z and the subgroup 6 1 of S 23 2 3 4

Part 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

14 5pts Write the base four number shown below in base ten 10

5 2 1 and y 2 4 6 find the vector projection of y onto x 1 10000

If 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

7 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

Let 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

Write 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

3 1 A 2 10 compute inverse matrix A 3 0 2 1 5

Let 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

Let 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

Problem 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

Let 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

of 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

1 point Calculate the area of the parallelogram determined by the following vectors Area 5 2 0 4

Let 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

ii 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

12 If B B and B are bases for R and if PA and Pa 3 then Pa B 2

Problem 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

Solve the equation for x 5x x 21 x 20 The solution set is

Find 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

1 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

c 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

Use 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

Use 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

Use 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

3 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

2 Using the inner product on R2 generated by matrix A 3 find u v for the vectors u 1 2 and v 2 4

The 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

for 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

1 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

3 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

For 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

and 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

c 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

nd 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

find 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

Matrix 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

Consider 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

Use 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

Let 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

Let 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

Use 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

A 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

MY 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

MY 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

nd 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

8 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

3 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

Let V 9 and B be the ordered basis B 1 H 5 Find the coordinates of the vector V relative to B