Matrices & Determinants Questions and Answers

Find the inverse of the affine transformation F x 2 x 2

Which graph best represents the solution of the inequality y22 x 11 27 SA ty Select one OA OB OC OD ty of 2 O

Find F dr for the given F and C 7 y7 x 3k and C is the helix x cos t y sin t z t for 0 t 3x 17 d F dr 113 x

10 points 8 a Let A 1 3 27 31 1 24 Find a basis for ColA

77 points 10 The matrix A 8 1 2 is symmetric and therefore orthogonally diagonalizable The characteristic equation of A is 2 5 2 16 0 Find an orthogonal matrix P that is a matrix P with orthonormal columns such that A PDP PDP

Find the solution set of the system of linear equations represented by the augmented matrix If there is no solution enter NO SOLUTION the system has an infinite number of solutions set x4 t and solve for X X2 and x3 in terms of t 1 2014 0 1 2 1 5 0 0 12 2 0 0 0 1 4 x1 X2 X3 X4

m 1 m 2 2 n Given two 3 matrices A 1 and E 1 1 11 2 1 0 values of m and n such that A E has an inverse 15 marks calculate A and A E Find the

2 pts Calculate the characteristic values and characteristic vectors of the following matrix and verify that it is equal to the determinant 3 4 21 A 0 1 2

0 0 0 0 0 Ix 11 12 OTO 0 1 1 0 0 0

m 1 1 m 3 Given two 3 matrices A 11 2 n and E 11 1 1 2 1 0 values of m and n such that A E has an inverse 15 marks calculate A and A El Fin

Use the matrices below to perform matrix multiplication B 4 po If the operation is undefined enter NA sin a 4 7 5 8 0 11 1 De Hint To create a matrix in Mobius click on the icon that has three rows of three dots it s nex the infinity symbol Specify the appropriate number of rows horizontal and columns vertical 00 D a 12 3 4 12 1 7 10 8 4 0

4 Find the inverses of the following matrices 1 1 1 0 0 10 1 1 7 100 2 0 23 1 1

Let and A 3 1 50 1 3 1 3 21 0 40 0 4 2 adj A a 24 8 16 b 40 2 4 4 8 20 40 71 28 19 30 Part a 5 points Use cofactor expansion along row 2 find det A Part a 4 points Find a and b

3 A 10 1 2 2 3 1 0 compute inverse matrix 4

or the matrix A below find a value of k so that A has two basic eigenvectors associated with the eigenvalue 2 2 3 3 9 7 0 2 k 6 0 0 3 2 00 0 2

Cosine of Angle of Rotation 3d The following matrix is the matrix of rotation Find the cosine of the angle of rotation 2 2 Q Answer 2 3 T 3 TO TOTO 2 T 2 3

Find the least squares solution of the system Ax b 12 H 1 1 3 1 X A b

ompute the determinant of the matrix A below by using row operations to transform A to an upper triangular matrix B Then express the determinant of A as a multiple k of the determinant of B and use ompute the determinant of A 9 2 10 6 18 5 26 16 18 4 24 22 36 10 48 23 000 0 0 0 000 ot A kdot B

Let A be the matrix of the quadratic form below It can be shown that the eigenvalues of A are 5 8 and 11 Find an orthogonal matrix P such that the change of variable x Py transforms x Ax into a quadratic form with no cross produc term Give P and the new quadratic form 8x 7x2 9x 4x x2 4x1x3 Which of the following matrices can be used in x Py to produce a quadratic form with no cross product term OA 1 2 2 2 2 0 1 The corresponding quadratic form is y Dy Simplify your answer B T 2 3 13 W N 1 1 3 3 2 2 1 3 3 2 www W N WIN 3 2 I 73 2 W N 3 C I 3 2 7 7 67 2 6 7 7 19 2 3 7 7 6 7 7

Express the general solution of Ax 8 24 a basis for null A b 4224 as x Xr cxn where x is the row space solution and Xn is X Ex 1 2 c Ex 5 1 b where A and

Find all distinct real or complex eigenvalues of A Then find the basic eigenvectors of A corresponding to each eigenvalue For each eigenvalue specify the number of basic eigenvectors corresponding to that eigenvalue then enter the eigenvalue followed by the basic eigenvectors corresponding to that eigenvalue A 1 5 2 5 Number of distinct eigenvalues 3 Number of Vectors 1 B Number of Vectors 1 0 0 Number of Vectors 1 0

Question 5 Compute A2011 if A Question 2 4 10 6 State the definition of similar matrices and prove that similar matrice have equal determinants and equal characteristic polynomials 02

Which of the following I A E 9 1 F D III I and II B II and III tement The set of all 2x2 matrices of the form with with the standard operations is a vector space The set of all 2x2 matrices of the form with the standard operations is a vector space The set of all 2x2 matrices of the form with the standard operations is a vector with space

1 point Let A and B be the following matrices Perform the following operations A B B A 5 2 22 3 3 A 1 7 B 9 8 8 2 3

1 point Find a non zero 2 x 2 matrix such that 4 8 8 16

Consider the initial value problem t C a Find the eigenvalue X an eigenvector 1 and a generalized eigenvector 2 for the coefficient matrix of this linear system HE v2 c Solve the original initial value problem y t 712 t 1 v1 b Find the most general real valued solution to the linear system of differential equations Use t as the independent variable in your answers N 0 c2

The system of linear equations has infinitely many solutions The general solution for the system is of the form of x a Av for ER vectors a and v in the box below where x a V x 2y 2z 3 x y 1 Express the solution in the above form and use Maple syntax to enter th a Note The vector b Q in Maple syntax should be entered as a b c

how can I use the moving and additivity principle to find area of the shaded triangle 4 inches 4 inches 5 inches A On your paper calculate the area of the shaded triangle inside the square below Explain clearly how you use the moving and additivity principles to find the area of the shaded triangle B In the blank type the area of the shaded region

4 MARKS is correct Let A 2 be an eigenvalue of an n x n matrix A Which of the following a 21 A 0 b 21 A is invertible c The rank of 21 A is strictly less than n e None of these d The eigenvector of A corresponding to 2 is zero

Q5 For what value of k does the following system of linear equations x k 1 y 5 k 1 x 9y 8k 1 have 1 An Infinite Number of Solutions 24 2 No Solution 3 A unique Solution

Given P 2 P P 9 a bc d P P d b calculate lim pr pte 9 Calculate p by indretion where a d 1 1 848 d bc Note In part a it is a d 1 not a d 1 saat afsny of AppOM The mod

nd a and d for the function f x a cos x d such that the graph of f matches the figure y T 2 3 4 5 6 7 8 9 10 12 13 14 B X

7 6 marks Consider matrices 4 6 3 A and B 2 a Find all 2 x 2 matrices C form a linear system for the entries of A and solve it which commute with A that is AC CA Hint

Consider 0 75 0 25 dx a 0 75x 0 25 da Hint draw the graph and interpret the integral as geometric area b Approximate the integral using a Riemann Sum with right endpoints and n 5 0 75 0 25 de Ar f xi f x f x f xi f x FC f f f EEE

2 Let A 1 2 2 1 23 11 2 2 3 H LAW 1 1 1 2 3 2 0 2 a Applying the QR factorization to solving the least squares problem Ax b gives the system 8 X b Use backsubstitution to solve the system in part a and find the least squares solution 2 1 3 0 x 20 30 1 1 and b 1 3 3 9 The QR factorization of the matrix A is given by 0 3

2 4 MARKS Find the standard matrix for the operator on R which contracts with factor 1 4 then reflects about the line y x 01 1 4 1 1 4 0 1 4 0 a A d A 1 4 abcde 0 1 A 1 4 0 e None of these c A 1 0

Given a b 40 A 0 3 0 10 2 Find the eigenvalues and bases for the eigenspaces of A Is A diagonalizable Justify your answer

1 point Let W be the set 1 3 Let W be the set 2 3 2 9 0 Determine if W is a basis for R and check the correct answer s below A W is a basis B W is not a basis because it does not span R C W is not a basis because it is linearly dependent 6 1 0 5 0 3 5 Determine if W is a basis for R and check the correct answer s below A W is a basis B W is not a basis because it is linearly dependent C W is not a basis because it does not span R

1 Compute the determinant of the following 5 x 5 matrix 1 1 1 1 1 2 A 1 1 1 3 3 4 2 23 24 3 3 33 34 4 2

1 point The matrix 4 8 4 4 0 2 2 has two real eigenvalues one of multiplicity 1 and one of multiplicity 2 Find the eigenvalues and a basis for each eigenspace The eigenvalue is The eigenvalue is and a basis for its associated eigenspace is and a basis for its associated eigenspace is

1 6 points Find the inverse matrix of each of the following matrices if it exis A 002 010 1 0 0 B 3500 1200 0001 0010 C 35 00 1 200 0 1 0 1 1 010

Define the linear transformation T R T u T v R by T x 6 6 A 249 12 1 8 Ax Find the images of u 2 4 5 and 7 a under T

7 LU V Ive 1 L 30 17 02 27 08 4 O 4100 3216 1 1 1 1 Azz for the system with out multibying Lti

1 point Consider the linear system a Find the eigenvalues and eigenvectors for the coefficient matrix A 1 y 3 i 04 H 4 G and 1 5 5 b Find the real valued solution to the initial value problem Use t as the independent variable in your answers 3 1 Y2 1 y 3y 232 5y 3y2 y 0 8 3 2 0 15

Esercizio 2 Let s consider the following matrix A 2 a Find a basis of the kernel of A b Find the characteristic polynomial and eigenvalues of A e Find the eigenspaces of A and their basis Is A diagonalizable d Does it exist a symmetric matrix similar to A you have to justify your ans

24 5 pts Show that the equation Ax b does not have a solution for all possible b where 1901031 202112190103A 00 0 A 2 2 and b b 3 1 2 by

The vectors B 2 B 1 18 5 3 form a basis Find a vector x given the B coordinates of x are

1 2 A 2 1 L2 2 27 2 FIND A 4A 11

Solve 18 II 2 1 2 608 3 17 12 X 13 30 51

Let A QR be the QR decomposition of A where A QR 1 2 0 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 0 0 1 2 0 1 2 Determine the minimum possible value of Ax b where 1 1 0 b 0 0 0 0