Question

Solve the following system of linear equations: 3x2−9x3 = −3 x1−2x2+x3 = 2 x2−3x3 = 0 If the system has no solution, demonstrate this by giving a row-echelon form of the augmented matrix for the system. If the system has infinitely many solutions, your answer may use expressions involving the parameters r, s, and t. You can resize a matrix (when appropriate) by clicking and dragging the bottom-right corner of the matrix.

Answer 1

The given homogeneous linear equations are 3x₂−9x₃=0 or, x₂−3x₃=0…(1),−3x₁−2x₂+x₃ = 0 or, 3x₁+2x₂-x₃ = 0…(2)and 2x₂−3x₃ = 0…(3).

The coefficient matrix of the given homogeneous system of linear equations is A (say) =

0	1	-3
3	2	-1
0	2	-3

To solve the given homogeneous system of linear equations, we will reduce A to its RREF as under:

Interchange the 1st row and the 2nd row

Multiply the 1st row by 1/3

Add -2 times the 2nd row to the 3rd row

Multiply the 3rd row by 1/3

Add 3 times the 3rd row to the 2nd row

Add 1/3 times the 3rd row to the 1st row

Add -2/3 times the 2nd row to the 1st row

Then the RREF of A is I₃.
Hence, the only solutionn to the given homogeneous system of linear equations is the trivial solution i.e. x₁=0, x₂ = 0 and x₃ = 0.