Consider the given reduced row echelon form of the augnented matrix correspondin
ID: 3138809 • Letter: C
Question
Consider the given reduced row echelon form of the augnented matrix corresponding to (20 points) some homogeneous system Ar0. 1 0 -2 0 2 0 0 0 0 1 3 0 Note: You do not have the matrix A, vou can't find it, and you don't need it. (a) Write the solution to the homogeneous system in parametric vector form. (b) Provide any two specific non-trivial solutions to the above homogeneous system. Refer to these as vi and . e that r = | 4 | is a solution to Air b. Note: You do not need the vector b. -6 Suppos ns to Ar B. (c) Using v and from your response to part (b), find two other solutio (d) Give a general solution to A-b in parametric vector form.Explanation / Answer
(a). Let X=(x,y,z,w,u)T.Then the equation AX=0 is equivalent to x-2z+2u=0 or, x=2z-2u, y+z = 0 or, y = -z, and w-3u = 0 or, w = 3u. Then X = (2z-2u, -z,z,3u,u)T = z(2,-1,1,0,0)T+u(-2,0,0,3,1)T = r(2,-1,1,0,0)T+ t(-2,0,0,3,1)T, where r,t are arbitrary real numbers.
(b). Let r = 0 and t = 1. Then v1 =(-2,0,0,3,1)T . Further, let r = 1 and t = 0. Then v2 = (2,-1,1,0,0)T.
(c ). If x = (1,0,4,-6,5)T is a solution to AX = b, then x+v1 and x+v2 are also solutions to AX = b.
(d). The general solution to AX = b is x + r(2,-1,1,0,0)T+ t(-2,0,0,3,1)T, where r,t are arbitrary real numbers.