Hey lukasaurus.

Your first question is spot on.

For the second one I confirmed the answer in Octave which is:

>> B

B =

1 -2 3

1 1 1

2 -1 4

>> det(B)

ans = 0

>> rref(B)

ans =

1.00000 0.00000 1.66667

0.00000 1.00000 -0.66667

0.00000 0.00000 0.00000

This confirms no unique solution and this command:

>> X = [2;2;4]

X =

2

2

4

>> A = [B,X]

A =

1 -2 3 2

1 1 1 2

2 -1 4 4

>> rref(A)

ans =

1.00000 0.00000 1.66667 2.00000

0.00000 1.00000 -0.66667 0.00000

0.00000 0.00000 0.00000 0.00000

confirms that infinitely many solutions only occurs when k = 2 just as you have mentioned.

If I set k = 1 for example Octave gives us:

>> X = [2;1;1]

X =

2

1

1

>> A = [B,X]

A =

1 -2 3 2

1 1 1 1

2 -1 4 1

>> rref(A)

ans =

1.00000 0.00000 1.66667 0.00000

0.00000 1.00000 -0.66667 0.00000

0.00000 0.00000 0.00000 1.00000

which is inconsistent (note the 1.0000).

So all your answers have been confirmed with a computer program.