Find vlaues of k so matrix has no solutions, what if u change the position of k?

[mr_coffee]

Hello everyone! I have a problem, it says Find hte values of k for which the matrix sysem Ax = b has no solutions. I got the answer for this one,
A =
6 2 2
-1 2 3
2 -6 k

b =
1
-3
0

I row reduced and ending up gettting:
6 3 0
0 -1 2
0 0 14-k
so k = 14, will give no solutions.

After that he says find:
Image space, nullspace, col space and row space, what are there dimensions, i also did this correctly.

But what happens if he changed the problem and said, Find the values of k for which the matrix system Ax = b has infitinatly many solutions or 1 unquie solution, then what would i do different? Thanks! I'm just trying to get all angles of this problem so i don't mess it up on the exam.

Also what if he puts k in the b vector insteed of the matrix A?
like
b =
-1
3
k

 

[matt grime]

Hello everyone! I have a problem, it says Find hte values of k for which the matrix sysem Ax = b has no solutions. I got the answer for this one,
A =
6 2 2
-1 2 3
2 -6 k

b =
1
-3
0

I row reduced and ending up gettting:
6 3 0
0 -1 2
0 0 14-k
so k = 14, will give no solutions.

After that he says find:
Image space, nullspace, col space and row space, what are there dimensions, i also did this correctly.

But what happens if he changed the problem and said, Find the values of k for which the matrix system Ax = b has infitinatly many solutions or 1 unquie solution, then what would i do different? Thanks! I'm just trying to get all angles of this problem so i don't mess it up on the exam.

that depends on A, doesn't it. If A is invertible (cols linearly independent) then there is exactly one unique solution of Ax=b. assume A is singular. then if there exists one solution then there are infinitely many, so look if b is in the image.

Also what if he puts k in the b vector insteed of the matrix A?
like
b =
-1
3
k

depends on the A, same comments apply.

 

[mr_coffee]

that depends on A, doesn't it. If A is invertible (cols linearly independent) then there is exactly one unique solution of Ax=b.

thanks for the responce, To see if A is invertiable, and to find the value of k in which there is exactly 1 unique solution, what would i do to the matrix, is what i was asking, like to find no solutions i row reduced until i got an expression of k, i got k -14 = 0;
so if k = 14, u got no solutions, say he changed the directions and said, find k so that there is 1 uqniue solution, then would it be like
k - 14 = ? that's what i don't get

 

[matt grime]

I told you the answer to that: if the cols are linearly independent then there is a unique solution, and the cols are linearly indepndent exactly when k-14 doesn'te equal zero. you found k-14=0 for a reason, remember. something that is not linearly dependent is linearly independent, it's a binary option. and you know linearly dependent is equivalent to k=14=0...

 

[mr_coffee]

ohhh my bad, thanks! everything is running together in my mind hah sorrry.