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Friday, October 31, 2008

Problem (1)

Solve the Flowing homogeneous system of linear equation by Gauss-Jordan elimination

2x – y – 3z = 0

-x +2y – 3z = 0

x + y + 4z = 0

Solution: -

The given system is

2x – y – 3z = 0

-x +2y – 3z = 0 System (1)

x + y + 4z = 0

Let us represent the three linear equation of the system (1) by L1, L2, and L3 respectively. Reduce the system to echelon form by elementary operation.

Apply L1ßà L2

x + y + 4z = 0

-x +2y – 3z = 0 System (2)

2x – y – 3z = 0

Apply L2 à L2 +L1 and L3à L3 – 2L1

Thus we obtain the equivalent system is

x + y + 4z = 0

3y + z = 0 System (3)

–3 y – 11z = 0

Now apply L3à L3 + L2

Thus we obtain the equivalent system is

x + y + 4z = 0

3y + z = 0 System (4)

10z = 0

System (4) which is the echelon form of the given system (1). It has three equations with three variables. So the system has only tribal solution. The solution of the given system is

(x,y,z) = (0,0,0)

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