Let A = and let x vector = Find the solution of the homogeneous equation Ax vext
ID: 3408650 • Letter: L
Question
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).Explanation / Answer
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
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