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The inverse matrix of A and B. The inverse matrix of A and B is [ − 3.2 2.2 − 2.8 2.2 − 1.2 1.8 − 2.8 1.8 − 2.2 ] , does not exist _ , respectively. Given: The matrices are A = [ 3 1 − 3 1 4 2 − 3 2 5 ] , B = [ 0 4 1 − 4 0 − 2 − 1 2 0 ] , u = [ 2 0 − 5 ] , v = [ 7 − 3 3 ] . Definition used: Inverse of a matrix: The inverse of the matrix is given by A − 1 = 1 det A a d j A . Calculation: Find the determinant of A . det A = | 3 1 − 3 1 4 2 − 3 2 5 | = 3 | 4 2 2 5 | − 1 | 1 2 − 3 5 | − 3 | 1 4 − 3 2 | = 3 ( 20 − 4 ) − 1 ( 5 + 6 ) − 3 ( 2 + 12 ) = 48 − 11 − 42 = − 5 Since the det A is non zero, the inverse of A exists. The cofactor matrix of A is given by, C o f a c t o r A = [ | 4 2 2 5 | − | 1 2 − 3 5 | | 1 4 − 3 2 | − | 1 − 3 2 5 | | 3 − 3 − 3 5 | − | 3 1 − 3 2 | | 1 − 3 4 2 | − | 3 − 3 1 2 | | 3 1 1 4 | ] = [ 16 − 11 14 − 11 6 − 9 14 − 9 11 ] The adjoint of A is the transpose of the cofactor of A . a d j A = [ C o f a c t o r A ] T = [ 16 − 11 14 − 11 6 − 9 14 − 9 11 ] T = [ 16 − 11 14 − 11 6 − 9 14 − 9 11 ] Thus, the inverse of the matrix is given by, A − 1 = 1 det A a d j A = 1 − 5 [ 16 − 11 14 − 11 6 − 9 14 − 9 11 ] = [ − 16 5 11 5 − 14 5 11 5 − 6 5 9 5 − 14 5 9 5 − 11 5 ] = [ − 3.2 2.2 − 2.8 2.2 − 1.2 1.8 − 2.8 1.8 − 2.2 ] Now, find the determinant of B . det B = | 0 4 1 − 4 0 − 2 − 1 2 0 | = 0 | 0 − 2 2 0 | − 4 | − 4 − 2 − 1 0 | + 1 | − 4 0 − 1 2 | = 0 − 4 ( − 2 ) + 1 ( − 8 ) = 8 − 8 = 0 Since the determinant becomes 0, the inverse does not exist. Thus, the inverse matrix of A and B is [ − 3.2 2.2 − 2.8 2.2 − 1.2 1.8 − 2.8 1.8 − 2.2 ] , does not exist _ , respectively.

The inverse matrix of A and B. The inverse matrix of A and B is [ − 3.2 2.2 − 2.8 2.2 − 1.2 1.8 − 2.8 1.8 − 2.2 ] , does not exist _ , respectively. Given: The matrices are A = [ 3 1 − 3 1 4 2 − 3 2 5 ] , B = [ 0 4 1 − 4 0 − 2 − 1 2 0 ] , u = [ 2 0 − 5 ] , v = [ 7 − 3 3 ] . Definition used: Inverse of a matrix: The inverse of the matrix is given by A − 1 = 1 det A a d j A . Calculation: Find the determinant of A . det A = | 3 1 − 3 1 4 2 − 3 2 5 | = 3 | 4 2 2 5 | − 1 | 1 2 − 3 5 | − 3 | 1 4 − 3 2 | = 3 ( 20 − 4 ) − 1 ( 5 + 6 ) − 3 ( 2 + 12 ) = 48 − 11 − 42 = − 5 Since the det A is non zero, the inverse of A exists. The cofactor matrix of A is given by, C o f a c t o r A = [ | 4 2 2 5 | − | 1 2 − 3 5 | | 1 4 − 3 2 | − | 1 − 3 2 5 | | 3 − 3 − 3 5 | − | 3 1 − 3 2 | | 1 − 3 4 2 | − | 3 − 3 1 2 | | 3 1 1 4 | ] = [ 16 − 11 14 − 11 6 − 9 14 − 9 11 ] The adjoint of A is the transpose of the cofactor of A . a d j A = [ C o f a c t o r A ] T = [ 16 − 11 14 − 11 6 − 9 14 − 9 11 ] T = [ 16 − 11 14 − 11 6 − 9 14 − 9 11 ] Thus, the inverse of the matrix is given by, A − 1 = 1 det A a d j A = 1 − 5 [ 16 − 11 14 − 11 6 − 9 14 − 9 11 ] = [ − 16 5 11 5 − 14 5 11 5 − 6 5 9 5 − 14 5 9 5 − 11 5 ] = [ − 3.2 2.2 − 2.8 2.2 − 1.2 1.8 − 2.8 1.8 − 2.2 ] Now, find the determinant of B . det B = | 0 4 1 − 4 0 − 2 − 1 2 0 | = 0 | 0 − 2 2 0 | − 4 | − 4 − 2 − 1 0 | + 1 | − 4 0 − 1 2 | = 0 − 4 ( − 2 ) + 1 ( − 8 ) = 8 − 8 = 0 Since the determinant becomes 0, the inverse does not exist. Thus, the inverse matrix of A and B is [ − 3.2 2.2 − 2.8 2.2 − 1.2 1.8 − 2.8 1.8 − 2.2 ] , does not exist _ , respectively.

ADVANCED ENGINEERING MATH W/ACCESS

ADVANCED ENGINEERING MATH W/ACCESS

10th Edition

ISBN: 9781119096023

Author: Kreyszig

Publisher: WILEY

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2. Consider the ODE u' = ƒ (u) = u² + r where r is a parameter that can take the values r = −1, −0.5, -0.1, 0.1. For each value of r: (a) Sketch ƒ(u) = u² + r and determine the equilibrium points. (b) Draw the phase line. (d) Determine the stability of the equilibrium points. (d) Plot the direction field and some sample solutions,i.e., u(t) (e) Describe how location of the equilibrium points and their stability change as you increase the parameter r. (f) Using the matlab program phaseline.m generate a solution for each value of r and the initial condition u(0) = 0.9. Print and turn in your result for r = −1. Do not forget to add a figure caption. (g) In the matlab program phaseline.m set the initial condition to u(0) = 1.1 and simulate the ode over the time interval t = [0, 10] for different values of r. What happens? Why? You do not need to turn in a plot for (g), just describe what happens.

True or False and why

10 5 Obtain by multiplying matrices the composite coordinate transformation of two transformations, first x' = (x + y√√2+2)/2 y' = z' (x√√2-2√2)/2 z = (-x+y√√2-2)/2 followed by x" = (x'√√2+z'√√2)/2 y" = (-x'y'√√2+2')/2 z" = (x'y'√√2-2')/2.

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ADVANCED ENGINEERING MATH W/ACCESS

Ch. 7.1 - Equality. Give reasons why the five matrices in...Ch. 7.1 - Double subscript notation. If you write the matrix...Ch. 7.1 - Sizes. What sizes do the matrices in Examples 1,...Ch. 7.1 - Main diagonal. What is the main diagonal of A in...Ch. 7.1 - Scalar multiplication. If A in Example 2 shows the...Ch. 7.1 - If a 12 × 12 matrix A shows the distances between...Ch. 7.1 - Addition of vectors. Can you add: A row and a...Ch. 7.1 - Let Find the following expressions, indicating...Ch. 7.1 - Let Find the following expressions, indicating...Ch. 7.1 - Let Find the following expressions, indicating...

Ch. 7.1 - Let Find the following expressions, indicating...Ch. 7.1 - Let Find the following expressions, indicating...Ch. 7.1 - Let Find the following expressions, indicating...Ch. 7.1 - Let Find the following expressions, indicating...Ch. 7.1 - Let Find the following expressions, indicating...Ch. 7.1 - Let Find the following expressions, indicating...Ch. 7.1 - Let Find the following expressions, indicating...Ch. 7.1 - Prob. 18P Ch. 7.1 - Prob. 19P Ch. 7.1 - TEAM PROJECT. Matrices for Networks. Matrices have...Ch. 7.2 - Multiplication. Why is multiplication of matrices...Ch. 7.2 - Square matrix. What form does a 3 × 3 matrix have...Ch. 7.2 - Product of vectors. Can every 3 × 3 matrix be...Ch. 7.2 - Skew-symmetric matrix. How many different entries...Ch. 7.2 - Same questions as in Prob. 4 for symmetric...Ch. 7.2 - Triangular matrix. If U1, U2 are upper triangular...Ch. 7.2 - Idempotent matrix, defined by A2 = A. Can you find...Ch. 7.2 - Nilpotent matrix, defined by Bm = 0 for some m....Ch. 7.2 - Transposition. Can you prove (10a)–(10c) for 3 × 3...Ch. 7.2 - Transposition. (a) Illustrate (10d) by simple...Ch. 7.2 - Let Showing all intermediate results, calculate...Ch. 7.2 - Let Showing all intermediate results, calculate...Ch. 7.2 - Let Showing all intermediate results, calculate...Ch. 7.2 - Let Showing all intermediate results, calculate...Ch. 7.2 - Let Showing all intermediate results, calculate...Ch. 7.2 - Let Showing all intermediate results, calculate...Ch. 7.2 - Let Showing all intermediate results, calculate...Ch. 7.2 - Let Showing all intermediate results, calculate...Ch. 7.2 - Let Showing all intermediate results, calculate...Ch. 7.2 - Let Showing all intermediate results, calculate...Ch. 7.2 - Prob. 21P Ch. 7.2 - Product. Write AB in Prob. 11 in terms of row and...Ch. 7.2 - Product. Calculate AB in Prob. 11 columnwise. See...Ch. 7.2 - Commutativity. Find all 2 × 2 matrices A = [ajk]...Ch. 7.2 - TEAM PROJECT. Symmetric and Skew-Symmetric...Ch. 7.2 - Production. In a production process, let N mean...Ch. 7.2 - Concert subscription. In a community of 100,000...Ch. 7.2 - Profit vector. Two factory outlets F1 and F2 in...Ch. 7.2 - TEAM PROJECT. Special Linear Transformations....Ch. 7.3 - Prob. 1P Ch. 7.3 - Solve the linear system given explicitly or by its...Ch. 7.3 - Solve the linear system given explicitly or by its...Ch. 7.3 - Solve the linear system given explicitly or by its...Ch. 7.3 - Prob. 5P Ch. 7.3 - Solve the linear system given explicitly or by its...Ch. 7.3 - Solve the linear system given explicitly or by its...Ch. 7.3 - Solve the linear system given explicitly or by its...Ch. 7.3 - Solve the linear system given explicitly or by its...Ch. 7.3 - Prob. 10P Ch. 7.3 - Prob. 11P Ch. 7.3 - Prob. 12P Ch. 7.3 - Prob. 13P Ch. 7.3 - Prob. 14P Ch. 7.3 - Prob. 15P Ch. 7.3 - Prob. 17P Ch. 7.3 - Prob. 18P Ch. 7.3 - Prob. 19P Ch. 7.3 - Prob. 20P Ch. 7.3 - Prob. 21P Ch. 7.3 - Prob. 22P Ch. 7.3 - Prob. 23P Ch. 7.3 - Prob. 24P Ch. 7.4 - Find the rank. Find a basis for the row space....Ch. 7.4 - Find the rank. Find a basis for the row space....Ch. 7.4 - Find the rank. Find a basis for the row space....Ch. 7.4 - Find the rank. Find a basis for the row space....Ch. 7.4 - Prob. 5P Ch. 7.4 - Find the rank. Find a basis for the row space....Ch. 7.4 - Find the rank. Find a basis for the row space....Ch. 7.4 - Prob. 8P Ch. 7.4 - Prob. 9P Ch. 7.4 - Prob. 10P Ch. 7.4 - Show the following: rank BTAT = rank AB. (Note the...Ch. 7.4 - Show the following: rank A = rank B does not imply...Ch. 7.4 - Prob. 14P Ch. 7.4 - Prob. 15P Ch. 7.4 - Prob. 16P Ch. 7.4 - Prob. 17P Ch. 7.4 - Prob. 18P Ch. 7.4 - Prob. 19P Ch. 7.4 - Prob. 20P Ch. 7.4 - Prob. 21P Ch. 7.4 - Prob. 22P Ch. 7.4 - Prob. 23P Ch. 7.4 - Prob. 24P Ch. 7.4 - Prob. 25P Ch. 7.4 - Prob. 26P Ch. 7.4 - Prob. 27P Ch. 7.4 - Prob. 28P Ch. 7.4 - Prob. 29P Ch. 7.4 - Prob. 30P Ch. 7.4 - Prob. 31P Ch. 7.4 - Prob. 32P Ch. 7.4 - Prob. 33P Ch. 7.4 - Prob. 34P Ch. 7.4 - Prob. 35P Ch. 7.7 - Prob. 1P Ch. 7.7 - Prob. 2P Ch. 7.7 - Prob. 3P Ch. 7.7 - Prob. 4P Ch. 7.7 - Prob. 5P Ch. 7.7 - Prob. 6P Ch. 7.7 - Showing the details, evaluate: Ch. 7.7 - Showing the details, evaluate: Ch. 7.7 - Showing the details, evaluate: Ch. 7.7 - Showing the details, evaluate: Ch. 7.7 - Showing the details, evaluate: Ch. 7.7 - Prob. 12P Ch. 7.7 - Prob. 13P Ch. 7.7 - Prob. 14P Ch. 7.7 - Prob. 15P Ch. 7.7 - Prob. 17P Ch. 7.7 - Prob. 18P Ch. 7.7 - Prob. 19P Ch. 7.7 - Prob. 21P Ch. 7.7 - Prob. 22P Ch. 7.7 - Prob. 23P Ch. 7.7 - Prob. 24P Ch. 7.7 - Prob. 25P Ch. 7.8 - Prob. 1P Ch. 7.8 - Prob. 2P Ch. 7.8 - Prob. 3P Ch. 7.8 - Prob. 4P Ch. 7.8 - Prob. 5P Ch. 7.8 - Prob. 6P Ch. 7.8 - Prob. 7P Ch. 7.8 - Prob. 8P Ch. 7.8 - Prob. 9P Ch. 7.8 - Prob. 10P Ch. 7.8 - Prob. 11P Ch. 7.8 - Prob. 12P Ch. 7.8 - Prob. 13P Ch. 7.8 - Prob. 14P Ch. 7.8 - Prob. 15P Ch. 7.8 - Prob. 16P Ch. 7.8 - Prob. 17P Ch. 7.8 - Prob. 18P Ch. 7.8 - Prob. 19P Ch. 7.8 - Prob. 20P Ch. 7.9 - Prob. 1P Ch. 7.9 - Prob. 2P Ch. 7.9 - Prob. 3P Ch. 7.9 - Prob. 4P Ch. 7.9 - Prob. 5P Ch. 7.9 - Prob. 6P Ch. 7.9 - Prob. 7P Ch. 7.9 - Prob. 8P Ch. 7.9 - Prob. 9P Ch. 7.9 - Prob. 10P Ch. 7.9 - Prob. 11P Ch. 7.9 - Prob. 12P Ch. 7.9 - Prob. 13P Ch. 7.9 - Prob. 14P Ch. 7.9 - Prob. 15P Ch. 7.9 - Prob. 16P Ch. 7.9 - Prob. 17P Ch. 7.9 - Prob. 18P Ch. 7.9 - Prob. 19P Ch. 7.9 - Prob. 20P Ch. 7.9 - Prob. 21P Ch. 7.9 - Prob. 22P Ch. 7.9 - Prob. 23P Ch. 7.9 - Prob. 24P Ch. 7.9 - Prob. 25P Ch. 7 - Prob. 1RQ Ch. 7 - Prob. 2RQ Ch. 7 - Prob. 3RQ Ch. 7 - Prob. 4RQ Ch. 7 - Prob. 5RQ Ch. 7 - Prob. 6RQ Ch. 7 - Prob. 7RQ Ch. 7 - Prob. 8RQ Ch. 7 - Prob. 9RQ Ch. 7 - Prob. 10RQ Ch. 7 - Prob. 11RQ Ch. 7 - Prob. 12RQ Ch. 7 - Prob. 13RQ Ch. 7 - Prob. 14RQ Ch. 7 - Prob. 15RQ Ch. 7 - Prob. 16RQ Ch. 7 - Prob. 17RQ Ch. 7 - Prob. 18RQ Ch. 7 - Prob. 19RQ Ch. 7 - Prob. 20RQ Ch. 7 - Prob. 21RQ Ch. 7 - Prob. 22RQ Ch. 7 - Prob. 23RQ Ch. 7 - Prob. 24RQ Ch. 7 - Prob. 25RQ Ch. 7 - Prob. 26RQ Ch. 7 - Prob. 27RQ Ch. 7 - Prob. 28RQ Ch. 7 - Prob. 29RQ Ch. 7 - Prob. 30RQ Ch. 7 - Prob. 31RQ Ch. 7 - Prob. 32RQ Ch. 7 - Prob. 33RQ Ch. 7 - Prob. 34RQ Ch. 7 - Prob. 35RQ

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