Question

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 a2l and Ibil 1 such that Hence find a lower triangular matrix L with unit diagonal elements such that A PLU where P- PP2 (b) The Hilbert matrix H3 of order three is given by 1 1/2 1/3 H3:=11/2 1/3 1/4 1/3 1/4 1/5 Develop the Cholesky method (factoring a real, symmetric, positive definite matrix) for H3. (c) Extend the result in (b) to the Hilbert matrix of order four, and summarize the al- gorithm in general.

Answer 1

Anguo LAR PA Ci) Q-5 A exist uation a)

tD0 2. O I ait0 Lj 0

Agci n. A- PLU 0 0 -26 249