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**1. Homework Statement**

Find k so that the system has exactly one solution.

[tex]\left{ \begin {array} {rcl} x - y + 2z = 1 \\ -x + y - z = 2 \\ x + ky + z = 3 \end {array} \right .[/tex]

**2. Homework Equations**

**3. The Attempt at a Solution**

Ok, so I create an augmented matrix here:

[tex]\left[ \begin {array} {ccc|r} 1 & -1 & 2 & 1 \\ -1 & 1 & -1 & 2 \\ 1 & k & 1 & 3 \end {array} \right][/tex]

Here I do the following: R2+R1 and R3-R1.

[tex]\left[ \begin {array} {ccc|r} 1 & -1 & 2 & 1 \\ 0 & 0 & 1 & 3 \\ 0 & k+1 & -1 & 2 \end {array} \right][/tex]

R2 <=> R3.

[tex]\left[ \begin {array} {ccc|r} 1 & -1 & 2 & 1 \\ 0 & k+1 & -1 & 2 \\ 0 & 0 & 1 & 3 \end {array} \right][/tex]

(1/k+1)R2.

[tex]\left[ \begin {array} {ccc|r} 1 & -1 & 2 & 1 \\ 0 & 1 & -1/(k+1) & 2/(k+1) \\ 0 & 0 & 1 & 3 \end {array} \right][/tex]

At this point, I'm stuck. How do I know which k will give the system exactly one solution?

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