Please help q05 Find an LU decomposition of the coefficient

Please help (q05)

Find an LU -decomposition of the coefficient matrix; then solve the given system by reducing the problem of solving the single system A_x = b to the problem of solving the two systems L_y = b and U_x = y having triangular coefficient matrices. [2 -2 -4 0 -2 6 -4 8 -2][x_1 x_2 x_3] = [0 -4 8]

Solution

To find LU decompositionof a matrix we will do following step

1. Use Gaussian Elimination to get the upper triangular matrix.

A=

1

to reduce in upper tringuler form use row operation R3=2R1+R3

THUS A becomes-

A=

AGAIN use row operation R3=2R2+R3 THUS A becomes-

A=

STEP 2-

NOW take identity matrix of order 3*3 and use the operation which are used to get upperv tringular matrix..

after applying R3=2R1+R3 AND R3 =2R2+R3 on I IT BECOMES-

L=

THUS A= LU

this is LU DECOMPOSITON of A. now to solve AX=B with the help of LU

consider AX=B

(LU)X=B implies that L(UX)=B

consider UX=Y which implies LY=B

FIRST we will solve LY=B and get Y then Solve UX =Y and get X..

Therefore LY=B

>>

let Y= [y1,y2,y3]

>>y1=0

y2=-4

2y1+2y2+y3=8.>>y3=16

now put Y IN UX=Y

>> let X=[x1,x2,x3]

after putting X in UX=Y we get x1=42,x2=26,x3=8

that is the ANSWER..

2	-2	-4
4	2	1
3	-1	1