Mathematical Methods in the Physical Sciences
3rd Edition
ISBN: 9780471198260
Author: Mary L. Boas
Publisher: Wiley, John & Sons, Incorporated
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Chapter 3.11, Problem 45P
Verify the details in the discussion of the matrices in
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Chapter 3 Solutions
Mathematical Methods in the Physical Sciences
Ch. 3.2 - The first equation in (2.6) written out in detail...Ch. 3.2 - Prob. 2PCh. 3.2 - For each of the following problems write and row...Ch. 3.2 - For each of the following problems write and row...Ch. 3.2 - For each of the following problems write and row...Ch. 3.2 - For each of the following problems write and row...Ch. 3.2 - For each of the following problems write and row...Ch. 3.2 - For each of the following problems write and row...Ch. 3.2 - For each of the following problems write and row...Ch. 3.2 - For each of the following problems write and row...
Ch. 3.2 - For each of the following problems write and row...Ch. 3.2 - For each of the following problems write and row...Ch. 3.2 - For each of the following problems write and row...Ch. 3.2 - For each of the following problems write and row...Ch. 3.2 - Find the rank of each of the following matrices....Ch. 3.2 - Find the rank of each of the following matrices....Ch. 3.2 - Find the rank of each of the following matrices....Ch. 3.2 - Find the rank of each of the following matrices....Ch. 3.3 - Evaluate the determinants in Problems 1 to 6 by...Ch. 3.3 - Evaluate the determinants in Problems 1 to 6 by...Ch. 3.3 - Evaluate the determinants in Problems 1 to 6 by...Ch. 3.3 - Evaluate the determinants in Problems 1 to 6 by...Ch. 3.3 - Evaluate the determinants in Problems 1 to 6 by...Ch. 3.3 - Evaluate the determinants in Problems 1 to 6 by...Ch. 3.3 - Prove the following by appropriate manipulations...Ch. 3.3 - Prob. 8PCh. 3.3 - Show without computation that the following...Ch. 3.3 - A determinant or a square matrix is called...Ch. 3.3 - In Problems 11 and 12 evaluate the determminants....Ch. 3.3 - In Problems 11 and 12 evaluate the determminants....Ch. 3.3 - Show that cos1012cos1012cos=cos3Ch. 3.3 - Show that the n-rowed determinant Hint: Expand...Ch. 3.3 - Use Cramers rule to solve Problem 2.3 and 2.11.Ch. 3.3 - In the following set of equations (from a quantum...Ch. 3.3 - Use Cramers rule to solve for x and t the Lorentz...Ch. 3.3 - Find z by Cramers rule:...Ch. 3.4 - Draw diagrams and prove (4.1).Ch. 3.4 - Given the vectors making the given angles With...Ch. 3.4 - Use vectors to prove the following theorems from...Ch. 3.4 - Use vectors to prove the following theorems from...Ch. 3.4 - Use vectors to prove the following theorems from...Ch. 3.4 - Use vectors to prove the following theorems from...Ch. 3.4 - Use vectors to prove the following theorems from...Ch. 3.4 - Use vectors to prove the following theorems from...Ch. 3.4 - Let A=2i+3j and B=4i4j. Show graphically, and find...Ch. 3.4 - If A+B =4j-i and A —B=i+3j, find A and B...Ch. 3.4 - Let 3i—j+4k, 7j—2k, i—3j+k be three vectors...Ch. 3.4 - Find the angle between the vectors A=2i+j2k and...Ch. 3.4 - If A = 4i-3k and B = —2i+2j— k, find the...Ch. 3.4 - Prob. 14PCh. 3.4 - Let A = 2i—j+2k. (a) Find a unit vector in the...Ch. 3.4 - Prob. 16PCh. 3.4 - Find three vectors (none of them parallel to a...Ch. 3.4 - Prob. 18PCh. 3.4 - Prob. 19PCh. 3.4 - Fine a vector perpendicular to both i+j and i-2k.Ch. 3.4 - Show that B|A|+A|B| and A|B|-B|A| are orthogonal.Ch. 3.4 - Square (A + B); interpret your result...Ch. 3.4 - If A = 2i—3j+ k and A • B = 0, does it follow...Ch. 3.4 - What is the value of (AB)2+(AB)2 ? Comment: This...Ch. 3.4 - Use vectors as in Problems 3 to 8, and also the...Ch. 3.4 - Use vectors as in Problems 3 to 8, and also the...Ch. 3.4 - Use vectors as in Problems 3 to 8, and also the...Ch. 3.4 - Use vectors as in Problems 3 to 8, and also the...Ch. 3.5 - In Problems 1 to 5, all lines are in the (x,y)...Ch. 3.5 - In Problems 1 to 5, all lines are in the (x,y)...Ch. 3.5 - In Problems 1 to 5, all lines are in the (x,y)...Ch. 3.5 - In Problems 1 to 5, all lines are in the (x,y)...Ch. 3.5 - In Problems 1 to 5, all lines are in the (x,y)...Ch. 3.5 - Prob. 6PCh. 3.5 - Prob. 7PCh. 3.5 - Prob. 8PCh. 3.5 - Prob. 9PCh. 3.5 - Prob. 10PCh. 3.5 - Prob. 11PCh. 3.5 - Prob. 12PCh. 3.5 - Find the symmetric equations (5.6) or (5.7) and...Ch. 3.5 - Prob. 14PCh. 3.5 - Prob. 15PCh. 3.5 - Prob. 16PCh. 3.5 - Prob. 17PCh. 3.5 - Prob. 18PCh. 3.5 - Prob. 19PCh. 3.5 - Find the symmetric equations (5.6) or (5.7) and...Ch. 3.5 - In Problems 21 to 23, find the angle between the...Ch. 3.5 - In Problems 21 to 23, find the angle between the...Ch. 3.5 - In Problems 21 to 23, find the angle between the...Ch. 3.5 - Find a point on both the planes (that is, on their...Ch. 3.5 - As in Problem 24, find the equations of the line...Ch. 3.5 - Prob. 26PCh. 3.5 - Find the equation of the plane through (2, 3,...Ch. 3.5 - Find the equation of the plane through (-4, -1, 2)...Ch. 3.5 - Find a point on the plane 2x — y — z = 13....Ch. 3.5 - Find the distance from the origin to the plane 3x...Ch. 3.5 - Find the distance from (-2, 4, 5) to the plane...Ch. 3.5 - Find the distance from (3, -1, 2) to the plane 5x...Ch. 3.5 - Findthe perpendicular distance between the two...Ch. 3.5 - Find the distance (perpendicular is understood)...Ch. 3.5 - Find the distance (2,5,1) to the line in Problem...Ch. 3.5 - Find the distance (3,2,5) to the line in Problem...Ch. 3.5 - Determine whether the lines x12=y+31=z43 and...Ch. 3.5 - Find the angle between the lines in Problem 37.Ch. 3.5 - In Problems 39 and 40, show that the given lines...Ch. 3.5 - In Problems 39 and 40, show that the given lines...Ch. 3.5 - In Problems 41 to 44, find the distance between...Ch. 3.5 - In Problems 41 to 44, find the distance between...Ch. 3.5 - In Problems 41 to 44, find the distance between...Ch. 3.5 - In Problems 41 to 44, find the distance between...Ch. 3.5 - A particle is traveling along the line (x — 3)/2...Ch. 3.6 - In Problems 1 to 3, find AB,BA,A+B,AB,A2,B2,5A,3B....Ch. 3.6 - In Problems 1 to 3, find AB,BA,A+B,AB,A2,B2,5A,3B....Ch. 3.6 - In Problems 1 to 3, find AB,BA,A+B,AB,A2,B2,5A,3B....Ch. 3.6 - Given the matrices A=23142105, B=241131,...Ch. 3.6 - Compute the product of each of the matrices in...Ch. 3.6 - The Pauli spin in quantum mechanics are...Ch. 3.6 - Find the matrix product 23142112 By evaluating...Ch. 3.6 - Show, by multiplying the matrices, that the...Ch. 3.6 - Find AB and BA given A=1236,B=10452. Observe that...Ch. 3.6 - Prob. 10PCh. 3.6 - Show that the unit matrix I has the property that...Ch. 3.6 - For the matrices in Example 3, verify that MM—1...Ch. 3.6 - In Problems 13 to 16, use (6.13) to find the...Ch. 3.6 - In Problems 13 to 16, use (6.13) to find the...Ch. 3.6 - In Problems 13 to 16, use (6.13) to find the...Ch. 3.6 - In Problems 13 to 16, use (6.13) to find the...Ch. 3.6 - Given the matrices A=111401420,B=101211212 (a)...Ch. 3.6 - Problem 17(b) is a special case of the general...Ch. 3.6 - In Problems 19 to 22, solve each set of equations...Ch. 3.6 - In Problems 19 to 22, solve each set of equations...Ch. 3.6 - In Problems 19 to 22, solve each set of equations...Ch. 3.6 - In Problems 19 to 22, solve each set of equations...Ch. 3.6 - Verify formula (6.13). Hint: Consider the product...Ch. 3.6 - Use the method of solving simultaneous equations...Ch. 3.6 - Verify (6.14) by multiplying the matrices and...Ch. 3.6 - In (6.14), let ==/2 and verify the result...Ch. 3.6 - Do Problem 26 if =/2,=/4.Ch. 3.6 - Verify the calculations in (6.15), (6.16), and...Ch. 3.6 - Show that if A and B are matrices which dont...Ch. 3.6 - For the Pauli spin matrix A in Problem 6, find the...Ch. 3.6 - Repeat Problem 30 for the Pauli spin matrix C in...Ch. 3.6 - For the Pauli spin matrix B in Problem 6, find eiB...Ch. 3.7 - Prob. 1PCh. 3.7 - Are the following linear functions? Prove your...Ch. 3.7 - Are the following linear functions? Prove your...Ch. 3.7 - Prob. 4PCh. 3.7 - Are the following linear vector functions? Prove...Ch. 3.7 - Are the following linear vector functions? Prove...Ch. 3.7 - Are the following operators linear? Definite...Ch. 3.7 - Are the following operators linear? Find the...Ch. 3.7 - Are the following operators linear? Find the...Ch. 3.7 - Are the following operators linear? Find the...Ch. 3.7 - Are the following operators linear? Find the...Ch. 3.7 - Are the following operators linear? Let D stand...Ch. 3.7 - Are the following operators linear? (a) As in...Ch. 3.7 - Are the following operators linear? Find the...Ch. 3.7 - Are the following operators linear? Find the...Ch. 3.7 - Are the following operators linear? Find the...Ch. 3.7 - Are the following operators linear? Find the...Ch. 3.7 - With the cross product of two vectors defined by...Ch. 3.7 - If multiply a complex number z=ri by ei, we get...Ch. 3.7 - Verify equations (7.13) using Figure 7.5. Hints:...Ch. 3.7 - Do the details Of Example 3 as follows: Verify...Ch. 3.7 - Let each of the following matrices represent an...Ch. 3.7 - Let each of the following matrices represent an...Ch. 3.7 - Let each of the following matrices represent an...Ch. 3.7 - Let each of the following matrices represent an...Ch. 3.7 - Let each of the following matrices represent an...Ch. 3.7 - Let each of the following matrices represent an...Ch. 3.7 - Write the matrices which produce a rotation about...Ch. 3.7 - Construct the matrix corresponding to a rotation...Ch. 3.7 - For the matrices G and K in (7.21), find the...Ch. 3.7 - To see a physical example of non-commuting...Ch. 3.7 - For each of the following matrices, find its...Ch. 3.7 - For each of the following matrices, find its...Ch. 3.7 - For each of the following matrices, find its...Ch. 3.7 - For each of the following matrices, find its...Ch. 3.8 - Write each of the vectors (8.1) as a linear...Ch. 3.8 - In Problems 2 to 4, find out whether the given...Ch. 3.8 - In Problems 2 to 4, find out whether the given...Ch. 3.8 - In Problems 2 to 4, find out whether the given...Ch. 3.8 - Show that any vector V in a plane can be written...Ch. 3.8 - Use Problem 5 to write V = 3i + 5j as a linear...Ch. 3.8 - As in Problem 6, write V = 4i-5j in terms of the...Ch. 3.8 - In Problems 8 to 15, use (8.5) to show that the...Ch. 3.8 - In Problems 8 to 15, use (8.5) to show that the...Ch. 3.8 - In Problems 8 to 15, use (8.5) to show that the...Ch. 3.8 - In Problems 8 to 15, use (8.5) to show that the...Ch. 3.8 - In Problems 8 to 15, use (8.5) to show that the...Ch. 3.8 - In Problems 8 to 15, use (8.5) to show that the...Ch. 3.8 - In Problems 8 to 15, use (8.5) to show that the...Ch. 3.8 - In Problems 8 to 15, use (8.5) to show that the...Ch. 3.8 - Prove that if the Wronskian (8.5) is not...Ch. 3.8 - In Problems 17 to 20, solve the sets of...Ch. 3.8 - In Problems 17 to 20, solve the sets of...Ch. 3.8 - In Problems 17 to 20, solve the sets of...Ch. 3.8 - In Problems 17 to 20, solve the sets of...Ch. 3.8 - Find a condition for four points in space to lie...Ch. 3.8 - Find a condition for three lines in a plane to...Ch. 3.8 - Using (8.9), find the values of such that the...Ch. 3.8 - Using (8.9), find the values of such that the...Ch. 3.8 - Using (8.9), find the values of such that the...Ch. 3.8 - For each of the following, write the solution in...Ch. 3.8 - For each of the following, write the solution in...Ch. 3.8 - For each of the following, write the solution in...Ch. 3.9 - Use index notation as in 9.9 to prove the second...Ch. 3.9 - Use index notation to prove the distributive law...Ch. 3.9 - Given the following matrix, find the transpose,...Ch. 3.9 - Repeat Problem 3 given A=02i1i20300.Ch. 3.9 - Show that the product AAT is a symmetric matrix.Ch. 3.9 - Give numerical examples of: a symmetric matrix; a...Ch. 3.9 - Write each of the items in the second column of...Ch. 3.9 - Prove that ABt=BtAt. Hint: see 9.10. Verify 9.11,...Ch. 3.9 - In 9.1 we have defined the adjoint of a matrix as...Ch. 3.9 - Show that if a matrix is orthogonal and its...Ch. 3.9 - Show that a real Hermitian matrix is symmetric....Ch. 3.9 - Show that the definition of a Hermitian matrix...Ch. 3.9 - Show that the following matrix is a unitary...Ch. 3.9 - Prob. 14PCh. 3.9 - Show that the Pauli spin matrices (Problem 6.6)...Ch. 3.9 - Let Cij=1i+jMij be the cofactor of element aij in...Ch. 3.9 - Show that if A and B are symmetric, then AB is not...Ch. 3.9 - If A and B are symmetric matrices, show that their...Ch. 3.9 - Prove that TrAB=TrBA. Hint: see proof of (9.13)....Ch. 3.9 - Show that the determinant of a unitary matrix is a...Ch. 3.9 - Show that the transpose of a sum of matrices is...Ch. 3.9 - Show that a unitary matrix is a normal matrix,...Ch. 3.9 - Show that the following matrices are Hermitian...Ch. 3.9 - Show that an orthogonal transformation preserves...Ch. 3.9 - Show that the inverse of an orthogonal matrix is...Ch. 3.10 - Find the distance between the points 4,1,2,7 and...Ch. 3.10 - For the given sets of vectors, find the dimension...Ch. 3.10 - (a) Find the cosines of the angles between pairs...Ch. 3.10 - For each given set of basis vectors, use the...Ch. 3.10 - By 10.6 and 10.7, find the norms of A and B and...Ch. 3.10 - Write out the proof of the Schwarz inequality 10.9...Ch. 3.10 - Show that, in n-dimensional space, any n+1 vectors...Ch. 3.10 - Show that two different sets of basis vectors for...Ch. 3.10 - Write equations 10.6 to 10.9 in matrix form as...Ch. 3.10 - Prove that A+BA+B. This is called the triangle...Ch. 3.11 - Verify 11.7. Also verify 11.12 and find the...Ch. 3.11 - Verify that the two eigenvectors in 11.8 are...Ch. 3.11 - If C is orthogonal and M is symmetric, show that...Ch. 3.11 - Find the inverse of the rotation matrix in 7.13;...Ch. 3.11 - Show that the C matrix in 11.10 does represent a...Ch. 3.11 - Show that if C is a matrix whose columns are the...Ch. 3.11 - Generalize Problem 6 to three dimensions; to n...Ch. 3.11 - Show that under the transformation 11.1, all...Ch. 3.11 - Show that detC1MC=detM. Hints: See 6.6. What is...Ch. 3.11 - Show that TrC1MC=TrM. Hint: see (9.13). Thus show...Ch. 3.11 - Find the inverse of the transformation...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Let each of the following matrices M describe a...Ch. 3.11 - Let each of the following matrices M describe a...Ch. 3.11 - Let each of the following matrices M describe a...Ch. 3.11 - Let each of the following matrices M describe a...Ch. 3.11 - Let each of the following matrices M describe a...Ch. 3.11 - Let each of the following matrices M describe a...Ch. 3.11 - Find the eigenvalues and eigenvectors of the real...Ch. 3.11 - By multiplying out M=CDC1 the diagonal matrix...Ch. 3.11 - The characteristic equation for a second-order...Ch. 3.11 - Verify the eigenvalues and eigenvectors of matrix...Ch. 3.11 - Starting with 11.23, obtain 11.24. Hints: Take the...Ch. 3.11 - Verify equation 11.25. Hint: Remember from Section...Ch. 3.11 - Write out the detailed proof of 11.27. Hint:...Ch. 3.11 - Verify the details as indicated in diagonalizing H...Ch. 3.11 - Verify that each of the following matrices is...Ch. 3.11 - Verify that each of the following matrices is...Ch. 3.11 - Verify that each of the following matrices is...Ch. 3.11 - Verify that each of the following matrices is...Ch. 3.11 - Verify the details in the discussion of the...Ch. 3.11 - We have seen that an orthogonal matrix with...Ch. 3.11 - Find a unitary matrix U which diagonalizes A in...Ch. 3.11 - Show that an orthogonal matrix M with all real...Ch. 3.11 - Verify the results for F in the discussion of...Ch. 3.11 - Show that the trace of a rotation matrix equals...Ch. 3.11 - Show that each of the following matrices is...Ch. 3.11 - Show that each of the following matrices is...Ch. 3.11 - Show that each of the following matrices is...Ch. 3.11 - Show that each of the following matrices is...Ch. 3.11 - Show that each of the following matrices is...Ch. 3.11 - Show that each of the following matrices is...Ch. 3.11 - Show that if D is a diagonal matrix, then Dn is...Ch. 3.11 - Note in Section 6 [see (6.15)] that, for the given...Ch. 3.11 - Repeat the last part of Problem 58 for the matrix...Ch. 3.11 - The Caley-Hamilton theorem states that A matrix...Ch. 3.11 - At the end of Section 9 we proved that if H is a...Ch. 3.11 - Show that if matrices F and G can be diagonalized...Ch. 3.12 - Verify that 12.2 multiplied out is 12.1.Ch. 3.12 - Find the equations of the following conics and...Ch. 3.12 - Find the equations of the following conics and...Ch. 3.12 - Find the equations of the following conics and...Ch. 3.12 - Find the equations of the following conics and...Ch. 3.12 - Find the equations of the following conics and...Ch. 3.12 - Find the equations of the following conics and...Ch. 3.12 - Carry through the details of Example 2 to find the...Ch. 3.12 - For Problems 2 to 7, find the rotation matrix C...Ch. 3.12 - Verify equations 12.13 and 12.14. Solve 12.15 to...Ch. 3.12 - Verify the details of Example 4, equations 12.18...Ch. 3.12 - Verify the details of Example 5, equations 12.26...Ch. 3.12 - Verify the details of Example 6, equations 12.37...Ch. 3.12 - Find the characteristic frequencies and the...Ch. 3.12 - Find the characteristic frequencies and the...Ch. 3.12 - Find the characteristic frequencies and the...Ch. 3.12 - Find the characteristic frequencies and the...Ch. 3.12 - Find the characteristic frequencies and the...Ch. 3.12 - Find the characteristic frequencies and the...Ch. 3.12 - Carry through the details of Example 7.Ch. 3.12 - Find the characteristic frequencies and the...Ch. 3.12 - Find the characteristic frequencies and the...Ch. 3.12 - Find the characteristic frequencies and the...Ch. 3.13 - Write the four rotation matrices for rotations of...Ch. 3.13 - Following the text discussion of the cyclic group...Ch. 3.13 - Prob. 3PCh. 3.13 - Show that the matrices...Ch. 3.13 - Consider the group of order 4 with unit element I...Ch. 3.13 - Consider the integers 0, 1, 2, 3 under addition...Ch. 3.13 - Consider the set of numbers 1, 3, 5, 7 with...Ch. 3.13 - Verify 13.3 and 13.4. Hints: For the rotation and...Ch. 3.13 - Show that any cyclic group is Abelian. Hint: Does...Ch. 3.13 - Prob. 10PCh. 3.13 - Do Problem 10 for a rectangle. Note that now only...Ch. 3.13 - Verify 13.5 and then also show that A, B are the...Ch. 3.13 - Using the discussion of simultaneous...Ch. 3.13 - Use the multiplication table you found in Problem...Ch. 3.13 - By Problem 13, you know that the matrices in...Ch. 3.13 - Do Problem 15 for the group of matrices you found...Ch. 3.13 - Verify that the sets listed in 13.7c are groups.Ch. 3.13 - Prob. 18PCh. 3.13 - Verify that the sets listed in 13.7e are groups....Ch. 3.13 - Is the set of all orthogonal 3-by-3 matrices with...Ch. 3.13 - Prob. 21PCh. 3.14 - Verify the statements indicated in Examples 1 to 5...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.15 - Show that if each element of one row (or column)...Ch. 3.15 - What is wrong with the following argument? If we...Ch. 3.15 - Find the equations of the line through the points...Ch. 3.15 - Given the line r=3ij+2i+j2kt: Find the equation of...Ch. 3.15 - Write the equations of a straight line through the...Ch. 3.15 - Derive the formula D=ax0+by0+cz0da2+b2+c2 for the...Ch. 3.15 - Given the matrices A, B, C below, find or mark as...Ch. 3.15 - Given A=102ii3010i, find AT,A,At,A1.Ch. 3.15 - The following matrix product is used in discussing...Ch. 3.15 - The following matrix product is used in discussing...Ch. 3.15 - There is a one-to-one correspondence between...Ch. 3.15 - The vectors A=aibj and B=ci+dj form two sides of a...Ch. 3.15 - The plane 2x+3y+6z=6 intersects the coordinate...Ch. 3.15 - In Problems 14 to 17, multiply matrices to find...Ch. 3.15 - In Problems 14 to 17, multiply matrices to find...Ch. 3.15 - In Problems 14 to 17, multiply matrices to find...Ch. 3.15 - In Problems 14 to 17, multiply matrices to find...Ch. 3.15 - Prob. 18MPCh. 3.15 - Find the eigenvalues and eigenvectors of the...Ch. 3.15 - Find the eigenvalues and eigenvectors of the...Ch. 3.15 - Find the eigenvalues and eigenvectors of the...Ch. 3.15 - Find the eigenvalues and eigenvectors of the...Ch. 3.15 - Find the eigenvalues and eigenvectors of the...Ch. 3.15 - Find the eigenvalues and eigenvectors of the...Ch. 3.15 - Find the C matrix which diagonalizes the matrix M...Ch. 3.15 - Repeat Problem 25 for Problem 19. Find the C...Ch. 3.15 - In Problems 27 to 30, rotate the given quadric...Ch. 3.15 - In Problems 27 to 30, rotate the given quadric...Ch. 3.15 - In Problems 27 to 30, rotate the given quadric...Ch. 3.15 - In Problems 27 to 30, rotate the given quadric...Ch. 3.15 - Find the characteristic vibration frequencies of a...Ch. 3.15 - Do Problem 31 if the spring constants are...Ch. 3.15 - Prove the Caley-Hamilton theorem (Problem 11.60)...Ch. 3.15 - In problems 6.30 and 6.31, you found the matrices...Ch. 3.15 - Show that a square matrix A has an inverse if and...Ch. 3.15 - Write the three 3 by 3 matrices for 180 rotations...Ch. 3.15 - Show that for a given irreducible representation...Ch. 3.15 - For a cyclic group, show that every element is a...
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- A hat contains slips of paper numbered 1 through 6. You draw two slips of paper at random from the hat,without replacing the first slip into the hat.(a) (5 points) Write out the sample space S for this experiment.(b) (5 points) Express the event E : {the sum of the numbers on the slips of paper is 4} as a subset of S.(c) (5 points) Find P(E)(d) (5 points) Let F = {the larger minus the smaller number is 0}. What is P(F )?(e) (5 points) Are E and F disjoint? Why or why not?(f) (5 points) Find P(E ∪ F )arrow_forwardEvery chatgpt give wrong answer Plz no chatgptarrow_forwardIn addition to the in-school milk supplement program, the nurse would like to increase the use of daily vitamin supplements for the children by visiting homes and educating about the merits of vitamins. She believes that currently, about 50% of families with school-age children give the children a daily megavitamin. She would like to increase this to 70%. She plans a two-group study, where one group serves as a control and the other group receives her visits. How many families should she expect to visit to have 80% power of detecting this difference? Assume that drop-out rate is 5%.arrow_forward
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- The average number of minutes Americans commute to work is 27.7 minutes (Sterling's Best Places, April 13, 2012). The average commute time in minutes for 48 cities are as follows: Click on the datafile logo to reference the data. DATA file Albuquerque 23.3 Jacksonville 26.2 Phoenix 28.3 Atlanta 28.3 Kansas City 23.4 Pittsburgh 25.0 Austin 24.6 Las Vegas 28.4 Portland 26.4 Baltimore 32.1 Little Rock 20.1 Providence 23.6 Boston 31.7 Los Angeles 32.2 Richmond 23.4 Charlotte 25.8 Louisville 21.4 Sacramento 25.8 Chicago 38.1 Memphis 23.8 Salt Lake City 20.2 Cincinnati 24.9 Miami 30.7 San Antonio 26.1 Cleveland 26.8 Milwaukee 24.8 San Diego 24.8 Columbus 23.4 Minneapolis 23.6 San Francisco 32.6 Dallas 28.5 Nashville 25.3 San Jose 28.5 Denver 28.1 New Orleans 31.7 Seattle 27.3 Detroit 29.3 New York 43.8 St. Louis 26.8 El Paso 24.4 Oklahoma City 22.0 Tucson 24.0 Fresno 23.0 Orlando 27.1 Tulsa 20.1 Indianapolis 24.8 Philadelphia 34.2 Washington, D.C. 32.8 a. What is the mean commute time for…arrow_forward(b) 313 dy dx -y= 10 sin(2x)y; y(x) = ex-5 cos(2x)arrow_forwardnd ave a ction and ave an 48. The domain of f y=f'(x) x 1 2 (= x<0 x<0 = f(x) possible. Group Activity In Exercises 49 and 50, do the following. (a) Find the absolute extrema of f and where they occur. (b) Find any points of inflection. (c) Sketch a possible graph of f. 49. f is continuous on [0,3] and satisfies the following. X 0 1 2 3 f 0 2 0 -2 f' 3 0 does not exist -3 f" 0 -1 does not exist 0 ve tes where X 0 < x <1 1< x <2 2arrow_forwardMorningstar tracks the total return for a large number of mutual funds. The following table shows the total return and the number of funds for four categories of mutual funds. Click on the datafile logo to reference the data. DATA file Type of Fund Domestic Equity Number of Funds Total Return (%) 9191 4.65 International Equity 2621 18.15 Hybrid 1419 2900 11.36 6.75 Specialty Stock a. Using the number of funds as weights, compute the weighted average total return for these mutual funds. (to 2 decimals) % b. Is there any difficulty associated with using the "number of funds" as the weights in computing the weighted average total return in part (a)? Discuss. What else might be used for weights? The input in the box below will not be graded, but may be reviewed and considered by your instructor. c. Suppose you invested $10,000 in this group of mutual funds and diversified the investment by placing $2000 in Domestic Equity funds, $4000 in International Equity funds, $3000 in Specialty Stock…arrow_forwardThe days to maturity for a sample of five money market funds are shown here. The dollar amounts invested in the funds are provided. Days to Maturity 20 Dollar Value ($ millions) 20 12 30 7 10 5 6 15 10 Use the weighted mean to determine the mean number of days to maturity for dollars invested in these five money market funds (to 1 decimal). daysarrow_forwardc. What are the first and third quartiles? First Quartiles (to 1 decimals) Third Quartiles (to 4 decimals) × ☑ Which companies spend the most money on advertising? Business Insider maintains a list of the top-spending companies. In 2014, Procter & Gamble spent more than any other company, a whopping $5 billion. In second place was Comcast, which spent $3.08 billion (Business Insider website, December 2014). The top 12 companies and the amount each spent on advertising in billions of dollars are as follows. Click on the datafile logo to reference the data. DATA file Company Procter & Gamble Comcast Advertising ($billions) $5.00 3.08 2.91 Company American Express General Motors Advertising ($billions) $2.19 2.15 ETET AT&T Ford Verizon L'Oreal 2.56 2.44 2.34 Toyota Fiat Chrysler Walt Disney Company J.P Morgan a. What is the mean amount spent on advertising? (to 2 decimals) 2.55 b. What is the median amount spent on advertising? (to 3 decimals) 2.09 1.97 1.96 1.88arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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