Question

7. (20 points) Let 0-1 5 3 A -2 34 2 -3-5 (a) ( 15 points) Solve the linear system Ax = b by Gaussian elimination and express the general solution in vector form. (b) (5 points) Write down the corresponding homogenous system Ax-0 explicitly and determine all non-trivial solutions from (a) without resolving the system

Answer 1

here system Ax=b is

345 531 123

augmented matrix is

0	-1	5	-3	8
1	-2	3	-4	5
2	-3	1	-5	2

convert into Reduced Row Eschelon Form...

Swapping row2 with row1

1	-2	3	-4	5
0	-1	5	-3	8
2	-3	1	-5	2

Add (-2 * row1) to row3

1	-2	3	-4	5
0	-1	5	-3	8
0	1	-5	3	-8

Divide row2 by -1

1	-2	3	-4	5
0	1	-5	3	-8
0	1	-5	3	-8

Add (-1 * row2) to row3

1	-2	3	-4	5
0	1	-5	3	-8
0	0	0	0	0

Add (2 * row2) to row1

1	0	-7	2	-11
0	1	-5	3	-8
0	0	0	0	0

reduced system is

0 1 -5 3 -8 u)

$x-7z+2w=-11...............................x=-11+7z-2w$

$y-5z+3w=-8..........................y=-8+5z-3w$

$z=z$.................free

$w=w$..................free

.

general solution is

23 7510 -500

.

.

for system Ax=0 ,,,,,,,,,, reduced vector b will be zero

hence general solution for system Ax=0 is

23 7510 2

3 7510