# Bx=b showing consistency

• Jul 3rd 2013, 05:55 PM
n22
Bx=b showing consistency
Show that the system of equations Bx=b is consistent for all bεℝ³

B=[1 2 0 1;1 0 1 -1;0 -1 3 -4]

; means row; This is a 3 by3 matrix
Often in answers I find that they say let ( x y z ) be an element of R^3 and then they put it in the matrix(on the Rhs as a solution) and do reduced row echelon...Is that necessary here? why is that necessary in other cases?
(its possible that I dont need to do RREs but lets say i did ..is this method correct?)
Anyway heres what I have done.
I would appreciate feedback.THanks.
Attachment 28722Attachment 28723

• Jul 3rd 2013, 07:14 PM
chiro
Re: Bx=b showing consistency
Hey n22.

Hint: Reduce the matrix and if you get a rank of less than 3 (i.e. zeroes in the first three columns for a row), then see if the fourth corresponding value is zero or non-zero.

The idea is that if 0x + 0y + 0z != 0, then you know that you have an inconsistent system.
• Jul 4th 2013, 06:16 AM
n22
Re: Bx=b showing consistency
Hi Chiro,
Finally I end up with this matrix: Attachment 28730

[x y z]=t[3 -2 -2] where t is some parameter which is an element of R..please not this is a column vector..if there is a parameter t doesnt that mean that there are an infinite number of solutions ?
What does this mean for the matrix???
i guess this means its consistent then ....since there is a solution for each leading one

Quote:

Originally Posted by chiro
Hey n22.

Hint: Reduce the matrix and if you get a rank of less than 3 (i.e. zeroes in the first three columns for a row), then see if the fourth corresponding value is zero or non-zero.

The idea is that if 0x + 0y + 0z != 0, then you know that you have an inconsistent system.

• Jul 4th 2013, 07:48 PM
chiro
Re: Bx=b showing consistency
Since you are reducing in terms of free parameters x, y, and z, then what did you get in terms of x,y,z for the final column? (You have all zeroes but they should be functions of x, y, and z).

If you want to check the answer, I suggest you get something like Maple or Mathematica (Maple is good for this problem).