Homework Answers
7. Let A [aij] be an n x n invertible tridiagonal matrix, that is aij= 0 if |i - j > 1. Compute the number of operat...
7. Let A [aij] be an n x n invertible tridiagonal matrix, that is aij= 0 if |i - j > 1. Compute the number of operations needed to solve the system Ax b by Gauss elimination without partial pivoting. (10 marks)
7. Let A [aij] be an n x n invertible tridiagonal matrix, that is aij= 0 if |i - j > 1. Compute the number of operations needed to solve the system Ax b by Gauss elimination without...
1.4. Let x[n] be a signal with x[n] = 0 for n < -2 and n...
1.4. Let x[n] be a signal with x[n] = 0 for n < -2 and n > 4. For each signal given below, determine the values of n for which it is guaranteed to be zero. (a) xịn - 3] (b) x[n+ 4] (c) x[-n] (d) x[-n+2] (e) x[-n-2] 1.5. Let x(t) be a signal with x(t) = 0 for t <3. For each signal given below, determine the values of t for which it is guaranteed to be zero....
a) Let I be the n x n identity matrix and let O be the n × n zero matrix
a) Let I be the n x n identity matrix and let O be the n × n zero matrix . Suppose A is an n × n matrix such that A3 = 0. Show that I + A is invertible and that (I + A)-1 = I – A+ A2. b) Let B and C be n x n matrices. Assume that the product BC is invertible. Show that B and C are both invertible.
1. (40pts) Let 8 >0 and hn: (8,2 - 8] -R be given by cos(n) hn...
1. (40pts) Let 8 >0 and hn: (8,2 - 8] -R be given by cos(n) hn (x) 72 Use Dirichlet's Test to show that the series hn converges uniformly on (8,27 - 8). That is, please solve the following problems: la. (10 pts) Let 9n (x) = . * € (8,27 - 8). Show that In - g uniformly, where g(x) = 0, for all 2 € (5,2 - 8) and 9n+1 () S (x). for all n e N...
4. Let A be an n x n matrix. Define the trace of A by the...
4. Let A be an n x n matrix. Define the trace of A by the formula tr(A) = 2 . In other words, the trace of a matrix is the sum of the diagonal entries of the matrix. It is known that for two n x n matrices A and B, the trace has the property that tr(AB) = tr(BA). Each of the following holds more generally, for n x n matrices A and B, but in the interest...
4. Let A and B be n x n such that B = 1-A and A2 = A. Show that AB BA = 0 4. Let A and B be n x n such that B 1-A and A2 = A. Show that AB-BA-0 4. Let A and B be n x n such that B = 1-A and...
4. Let A and B be n x n such that B = 1-A and A2 = A. Show that AB BA = 0 4. Let A and B be n x n such that B 1-A and A2 = A. Show that AB-BA-0
4. Let A and B be n x n such that B = 1-A and A2 = A. Show that AB BA = 0
4. Let A and B be n x n such that B...
a. Let B be an n x n Orthogonal matrix, that is B^-1 = B^T, and...
a. Let B be an n x n Orthogonal matrix, that is B^-1 = B^T, and let A be an n x n skew-symmetric matrix. Simplify A(A^2(BA)^-1)^T b. Let A be a square matrix such that A^3 = 0. A is then called a nilpotent matrix. Define another matrix B by the expression B = I - A; Show that B is invertible and that its inverse is I + A + A^2 c. Let B = (-2,0,0 ; 0,0,0...
(16). Determine the determinant of the following n x n matrix: 2 3 II 2 3...
(16). Determine the determinant of the following n x n matrix: 2 3 II 2 3 0 3 00 9 (17). If A= then A= 9 3 7 2 1 (18). Let A= 1 2 If x= is an eigenvector of A-1, then k = 1 2 (19). Let A € R3x3 and det(A - 1) = det(A + 1) = det(A - 21) = 0. Then det(A) = 1 3 3 2 (20). The rank of matrix A =...
Problem 2 Let A be an n x n matrix which is not 0 but A-0...
Problem 2 Let A be an n x n matrix which is not 0 but A-0 Let I be the identity matrix. a) (10 Points) Show that A is not diagonalizable. b) (5 Points) Show that A is not invertible. e) (5 Points) Show that I-A is invertible and find its inverse.
1. For the 5 x 5 tridiagonal matrix 2 0 0 0 -1 2 -1 0 0 T=10-1 2-1 0 0 0 -1 2-1 0 0 02 use Sturm ...
1. For the 5 x 5 tridiagonal matrix 2 0 0 0 -1 2 -1 0 0 T=10-1 2-1 0 0 0 -1 2-1 0 0 02 use Sturm sequences with bisection to find all of the eigenvalues in the interval [0, 1.5 to one significant digit
1. For the 5 x 5 tridiagonal matrix 2 0 0 0 -1 2 -1 0 0 T=10-1 2-1 0 0 0 -1 2-1 0 0 02 use Sturm sequences with bisection to...
7. Let A [aij] be an n x n invertible tridiagonal matrix, that is aij= 0 if |i - j > 1. Compute the number of operations needed to solve the system Ax b by Gauss elimination without partial pivoting. (10 marks)
7. Let A [aij] be an n x n invertible tridiagonal matrix, that is aij= 0 if |i - j > 1. Compute the number of operations needed to solve the system Ax b by Gauss elimination without...
1.4. Let x[n] be a signal with x[n] = 0 for n < -2 and n > 4. For each signal given below, determine the values of n for which it is guaranteed to be zero. (a) xịn - 3] (b) x[n+ 4] (c) x[-n] (d) x[-n+2] (e) x[-n-2] 1.5. Let x(t) be a signal with x(t) = 0 for t <3. For each signal given below, determine the values of t for which it is guaranteed to be zero....
a) Let I be the n x n identity matrix and let O be the n × n zero matrix . Suppose A is an n × n matrix such that A3 = 0. Show that I + A is invertible and that (I + A)-1 = I – A+ A2. b) Let B and C be n x n matrices. Assume that the product BC is invertible. Show that B and C are both invertible.
1. (40pts) Let 8 >0 and hn: (8,2 - 8] -R be given by cos(n) hn (x) 72 Use Dirichlet's Test to show that the series hn converges uniformly on (8,27 - 8). That is, please solve the following problems: la. (10 pts) Let 9n (x) = . * € (8,27 - 8). Show that In - g uniformly, where g(x) = 0, for all 2 € (5,2 - 8) and 9n+1 () S (x). for all n e N...
4. Let A be an n x n matrix. Define the trace of A by the formula tr(A) = 2 . In other words, the trace of a matrix is the sum of the diagonal entries of the matrix. It is known that for two n x n matrices A and B, the trace has the property that tr(AB) = tr(BA). Each of the following holds more generally, for n x n matrices A and B, but in the interest...
4. Let A and B be n x n such that B = 1-A and A2 = A. Show that AB BA = 0 4. Let A and B be n x n such that B 1-A and A2 = A. Show that AB-BA-0
4. Let A and B be n x n such that B = 1-A and A2 = A. Show that AB BA = 0
4. Let A and B be n x n such that B...
(16). Determine the determinant of the following n x n matrix: 2 3 II 2 3 0 3 00 9 (17). If A= then A= 9 3 7 2 1 (18). Let A= 1 2 If x= is an eigenvector of A-1, then k = 1 2 (19). Let A € R3x3 and det(A - 1) = det(A + 1) = det(A - 21) = 0. Then det(A) = 1 3 3 2 (20). The rank of matrix A =...
Problem 2 Let A be an n x n matrix which is not 0 but A-0 Let I be the identity matrix. a) (10 Points) Show that A is not diagonalizable. b) (5 Points) Show that A is not invertible. e) (5 Points) Show that I-A is invertible and find its inverse.
1. For the 5 x 5 tridiagonal matrix 2 0 0 0 -1 2 -1 0 0 T=10-1 2-1 0 0 0 -1 2-1 0 0 02 use Sturm sequences with bisection to find all of the eigenvalues in the interval [0, 1.5 to one significant digit
1. For the 5 x 5 tridiagonal matrix 2 0 0 0 -1 2 -1 0 0 T=10-1 2-1 0 0 0 -1 2-1 0 0 02 use Sturm sequences with bisection to...
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