Let A and let x vector Find the solution of the homogeneou

Let A = and let x vector = Find the solution of the homogeneous equation Ax vextor = 0 vector and state whether the equation has a nontrivial solution or a trivial solution. If the system has a nontrivial solution, write it in parametric vector form. Let b vector = Describe all solutions of the equation A x vector = b vector in parametric vector form. Recall that the solution to the equation A x vector = 0 vector is called a homogeneous solution. Identify (label) the homogeneous solution and particular solution in part (b).

Solution

A =     1       2        -3

           2       1        -3

          -1       1         0

X = X1

        X2

        X3

    a)

Ax = 0

we multiply matrix A with matrix x and put it equal to zero

1x₁ +2x₂ -3x₃                      0

2x₁ +1x₂ -3x₃        =      0

-1x₁ +1x₂ +0x₃                  0

we get

1x₁ +2x₂ -3x₃  = 0            (1)

2x₁ +1x₂ -3x₃ = 0             (2)

-1x₁ +1x₂       = 0              (3)   

now we have 3 equations

we can solve for x1,x2 and x3

eq(1) -eq(2) we get

-1x₁ +1x₂       = 0

we get eq(3) so we cannot solve this

x₁ = t

from (3) we have

x₂ = t

from (2)

3t -3x₃ =0

x₃ = t

x1            1     

x2    = t ( 1 )

x3            1

b)

A =     1       2        -3

           2       1        -3

          -1       1         0

X = X1

        X2

        X3

   and

B= 5

     13

     -8

ax = b

1x₁ +2x₂ -3x₃  = 5            (1)

2x₁ +1x₂ -3x₃ = 13            (2)

-1x₁ +1x₂       = -8             (3)   

we cannot sove these

therefore from (3)

x₂= -8+1x₁   let x₁ = t

x₂ = - 8 + t      (4)

x₂ = t - 8

from (2)

2x₁ +1x₂ -3x₃ = 13

2t + (t - 8) -3x₃ = 13

3t - 8 -3x₃ =13

-3x₃ = 13+8-3t

3x₃ = 3t - 21

x₃ = t - 7

x1                 1            0

x2      = t * ( 1)   - 1 (8)

x3                  1            7