1 2 -1 0 3 3 -6 9 0 2 0 10 -6 6 1 Use Gaussian elimination with partial pivoting to do PT LU decomposition as follows: (1). Express U as U = MPMPA and determine the matrices M's and P's that implement Gaussian elimination with partial pivoting. (2) Express A as A= PT LU and determine P and L.
3. Given the matrix [ -1 2 -1] A= 3 2 1 10 10 1 Following steps (a)(b) to obtain the LU decomposition of the matrix A with partial piv- oting (a) Apply the Gaussian elimination method with partial pivoting to obtain an upper trian- gular matrix U. Record the corresponding permutation matrix for each pivoting step, and the numbers lik used to eliminate the zeros in column k. (b) Based on (a), express the matrices P, L and U...
2. (a) Let A be the matrix A -4 21 8 -40 Write down the 3 x 3 permutation matrix P such that PA interchanges the 1st and 3rd rows of A. Find the inverse of P. Use Gaussian elimination with partial pivoting to find an upper triangular matrix U, permutation matrices Pi and P2 and lower triangular matrices Mi and M2 of the form 1 0 0 Mi-1A1 10 a2 0 1 M2 0 0 0 b1 with ail...
ALTSIS AND NUMERICAL ANALYSIS 2. (a) Let A be the matrix 2 -115 8-4 Write down the 3 x 3 permutation matrix P such that PA interchanges the 1st and 3rd rows of A. Find the inverse of P Use Gaussian elimination with partial pivoting to find an upper triangular matix U, permutation matrices Pi and P2 and lower triangular matrices M and M2 of the form 1 0 0 0 1 1 0 0 0 bi 1 with land...
Below are the results of each step in the transformation of a matrix to row reduced echelon form, using Gaussian elimination. These are the same steps involved in the decomposition PALU - 2 2 0 6 AP 1 1 2 0 0 0 2 -3 1 3 3 3 1 3 3 3 0 2 3 0 0 4 2 2 A o 2 3 0 A0 0 2 -13 0 2-1 0 0 0 -2 0 0 2 -3...
1. [A] is the coefficient matrix for [Aj[X]-(C. 12-10 16 A-16 9 24 12 8 At the end of forward elimination steps of Gaussian Elimination method with partial pivoting, the coefficient matrix looks like 0 0 by a) bs is most nearly (circle correct response) [10 pts.] A. -2.0298 B. 1.4167 C. 12.000 D. 22.667 b) This is a consistent/inconsistent system. (circle correct response) (5 points) A square matrix [A] is upper triangular if (circle correct response) |5 points (A)...
Problem X. Take the method for finding the inverse of a given n x n matrix A -a by straightforward Gauss (or Jordan) elimination (Problem 7 is a particular case for n 3). First you write down the augmented matrix A and apply the Gauss process to this as discussed in class: A-la2,1 a2,2 a2,n : an,1 an,2 .. an.n 0 0 1 3. Derive the Jordan elimination algorithm without pivoting for the augmented matrix in terms of a triple...
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. For the following two systems of linear equations answer the questions 4 + x + 2xy + 2x - 6 3x + 2x + 3x3 + 3x = 11 2x + 2x + 3.5+ 2x- 9 2x + 2x+4x3+5x - 13 3x, +2, +4x3+4x-13 3x+3x+3x2+2x, -11 (1) Solve the above systems of linear equations using naive Gauss elimination (b) solve the above systems of linear equations using Gauss elimination with partial pivoting . Axb 2. For the following matrix...
(911 (1) (a) Recall that a square matrix A has an LU decomposition if we can write it as the product A = LU of a lower triangular matrix and an upper triangular matrix. Show that the matrix 0 1 21 A= 3 4 5 (6 7 9] does not have an LU decomposition 0 0 Uji U12 U13 O 1 2 Il 21 l22 0 0 U22 U23 = 3 4 5 (131 132 133 0 0 U33 6...