Have you noticed that the matrix A is singular?
Have you looked at the row reduction of
?
I am having trouble answering this hw problem on section 1.4 in my linear algebra book by David C. Lay the updated 3rd edition. Anyhow here's the question
Let A= [2 -6(vertical column), -1 3 (vertical also) ] and b= [b1, b2] (vertical column)
Show that the equation Ax=b does not have a solution for all possible b, and describe the set of all b for which Ax=b does have a solution.
Thanks!
this is what I had for the reduction, I am not sure if it's correct...
column 1 [1 0] column 2 [-1/2 0] column 3 [b1/2 b2/3]
the answer in the back of the book gave the following response:
The equation Ax=b is consistent when 3b1+b2 is nonzero.(I need to show my work to show why it's not consistent). The set of b for which the equation is consistent is a line through the origin- the set of all points (b1,b2) satisfying b2=-3b1.
I don't know what any of this means. Could please explain?
Now to answer your question. This is quite easy.
Did you row-reduce the matrix?
So, we have the following (I'll use Maple syntax for showing the Matrix; if someone could convert it to "pretty print" I'd appreciate it):
Let A = [[2, -1],[-6, 3]] and let b = [[b_1], [b_2]]
By row reducing the AUGMENTED matrix of the system, we have the following:
[[2, -1, b_1],[-6, 3, b_2]] ~ (Multiply the top row by -3 and add to 2nd row):
[[2, -1, b_1],[0, 0, 3b_1 + b_2]]
Now, we can see that there is a PIVOT in the augmented column. Also, we see that if:
3b_1 + b_2 != 0, then Ax = b does NOT have a solution.
Further, we note that for all b such that 3b_1 + b_2 = 0, then the system DOES have a solution (consistent).
Does this make sense?
Here you go:
this means that in the 2 x 1 solution matrix x, and for some parameter . This means that . Now, if is anything other than 0, the matrix is inconsistent, since it would mean that something other than zero, which makes no sense! So we set , so . If we plot this solution on a set of axis, we would get a line through the origin, since would be analagous to and would be analagous to , we would essentially be plotting the line , which is a line through the origin.
I'm not that good at coding structures like matrices in LaTex, I just stole Plato's code and modified it
EDIT: In retrospect i think to say that is incorrect, i was trying to make some other connection and somehow confused myself along the way. pretty much everything i said after that is correct though. that is if then it would mean something other than zero ... and the argument follows from there