Homework Answers
Exercise 1. (a) Find the inverse of the matrix 0 0 1/2 A= 01/ 31 1/5...
Exercise 1. (a) Find the inverse of the matrix 0 0 1/2 A= 01/ 31 1/5 1 0 (b) Let N be a nxn matrix with N2 = 0. Show (I. - N)-1 = IA+N. (Hint: Use the definition of the inverse.)
110 In Problem 5.49, a least squares inverse was found for the matrix A = ſi...
110 In Problem 5.49, a least squares inverse was found for the matrix A = ſi -1 -2 -2 4 3 [ 1 1 -3 17 -2. 11 (a) Use this least squares inverse to show that the system of equations Az = C is inconsistent, where c' = (2,1,5). (b) Find a least squares solution. (c) Compute the sum of squared errors for a least squares solution to this system of equations.
ſi 0 1 37 14 00 11 1. Compute the determinant of 10 4 11 5...
ſi 0 1 37 14 00 11 1. Compute the determinant of 10 4 11 5 using cofactors. Show your work. 12 0 1 2
4. (3 points) Let ſi 2 1] A= 0 4 3 [1 2 2 Compute the...
4. (3 points) Let ſi 2 1] A= 0 4 3 [1 2 2 Compute the third column of A-1 by solving the equation Ax = es, where ez = 0 Hint: Use Cramar's rule to solve the equation, noticing that the third column of A-' is given by the solution of the above equation. In fact there is nothing special about A-1, the third column of any 3 x 3 matrix B is given by the product Bez. Can...
3 2 0 3. Compute the product 0 01-1 0 013 4. If the matrix A...
3 2 0 3. Compute the product 0 01-1 0 013 4. If the matrix A from the previous problem represents a linear transformation T, determine: (a.) Is the mapping onto (b.) Is the mapping one to one (c.) Is the mapping homomorphic (d.) Is the mapping isomorphic (e.) What is the range space? The rank? (f) What is the null space? The nullity? (g.) Does this transformation preserve magnitude? 5. (a.) What is AT, the transpose of the matrix...
Question 3) (8 points) Consider the following matrix: A= ſi 4 0 0 28 3 12...
Question 3) (8 points) Consider the following matrix: A= ſi 4 0 0 28 3 12 2 11 -5 5 6 0 8 1 (a) Find a basis for the Rowspace(A). Then state the dimension of the Rowspace(A). (b) Find a basis for the Colspace(A). Then state the dimension of the Colspace(A). (e) Find a basis for the Nullspace(A). Then state the dimension of the Nullspace(A). (d) State and confirm the Rank-Nullity Theorem for this matrix.
The eigenvalues of the symmetric matrix A= ſi 8 41 8 1 -4 are 11 =...
The eigenvalues of the symmetric matrix A= ſi 8 41 8 1 -4 are 11 = 9 and 12 = -9. 14 -4 7 | Find an orthogonal diagonalization of A. Find the characteristic polynomial of A.
Use cofactors to compute the inverse of the following matrix: A:1 -2 1. し0-12
Use cofactors to compute the inverse of the following matrix: A:1 -2 1. し0-12
Use Gauss elimination, compute the determinant of the matrix o 0 2 0-1 4 4 5...
Use Gauss elimination, compute the determinant of the matrix o 0 2 0-1 4 4 5 1 2 0 0 7 2 5 -1 5 6 5 0 -1 5 0 4 8
points 5. Find the inverse of the following matrix: 10 -1] -4 1 3 2 0...
points 5. Find the inverse of the following matrix: 10 -1] -4 1 3 2 0 3 | 1
Exercise 1. (a) Find the inverse of the matrix 0 0 1/2 A= 01/ 31 1/5 1 0 (b) Let N be a nxn matrix with N2 = 0. Show (I. - N)-1 = IA+N. (Hint: Use the definition of the inverse.)
110 In Problem 5.49, a least squares inverse was found for the matrix A = ſi -1 -2 -2 4 3 [ 1 1 -3 17 -2. 11 (a) Use this least squares inverse to show that the system of equations Az = C is inconsistent, where c' = (2,1,5). (b) Find a least squares solution. (c) Compute the sum of squared errors for a least squares solution to this system of equations.
ſi 0 1 37 14 00 11 1. Compute the determinant of 10 4 11 5 using cofactors. Show your work. 12 0 1 2
4. (3 points) Let ſi 2 1] A= 0 4 3 [1 2 2 Compute the third column of A-1 by solving the equation Ax = es, where ez = 0 Hint: Use Cramar's rule to solve the equation, noticing that the third column of A-' is given by the solution of the above equation. In fact there is nothing special about A-1, the third column of any 3 x 3 matrix B is given by the product Bez. Can...
3 2 0 3. Compute the product 0 01-1 0 013 4. If the matrix A from the previous problem represents a linear transformation T, determine: (a.) Is the mapping onto (b.) Is the mapping one to one (c.) Is the mapping homomorphic (d.) Is the mapping isomorphic (e.) What is the range space? The rank? (f) What is the null space? The nullity? (g.) Does this transformation preserve magnitude? 5. (a.) What is AT, the transpose of the matrix...
Question 3) (8 points) Consider the following matrix: A= ſi 4 0 0 28 3 12 2 11 -5 5 6 0 8 1 (a) Find a basis for the Rowspace(A). Then state the dimension of the Rowspace(A). (b) Find a basis for the Colspace(A). Then state the dimension of the Colspace(A). (e) Find a basis for the Nullspace(A). Then state the dimension of the Nullspace(A). (d) State and confirm the Rank-Nullity Theorem for this matrix.
The eigenvalues of the symmetric matrix A= ſi 8 41 8 1 -4 are 11 = 9 and 12 = -9. 14 -4 7 | Find an orthogonal diagonalization of A. Find the characteristic polynomial of A.
Use cofactors to compute the inverse of the following matrix: A:1 -2 1. し0-12
Use Gauss elimination, compute the determinant of the matrix o 0 2 0-1 4 4 5 1 2 0 0 7 2 5 -1 5 6 5 0 -1 5 0 4 8
points 5. Find the inverse of the following matrix: 10 -1] -4 1 3 2 0 3 | 1
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