(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find all [itex]r[/itex] so that the matrix [itex]\begin{bmatrix} 2 & 4 & 2 \\ 1 & r & 3 \\ 1 & 2 & 1\end{bmatrix}[/itex] is inversable.

2. Relevant equations

Gauss-Jordan elimination on a matrix augmented with an identity matrix.

3. The attempt at a solution

[itex]\begin{bmatrix} 2 & 4 & 2 & | & 1 & 0 & 0 \\ 1 & r & 3 & | & 0 & 1 & 0 \\ 1 & 2 & 1 & | & 0 & 0 & 1\end{bmatrix}[/itex]

(1/2)R1

[itex]\begin{bmatrix} 1 & 2 & 1 & | & \frac{1}{2} & 0 & 0 \\ 1 & r & 3 & | & 0 & 1 & 0 \\ 1 & 2 & 1 & | & 0 & 0 & 1\end{bmatrix}[/itex]

R2 - R1

R3 - R1

[itex]\begin{bmatrix} 1 & 2 & 1 & | &\frac{1}{2} & 0 & 0 \\ 0 & r - 2 & 2 & | &-\frac{1}{2} & 1 & 0 \\ 0 & 0 & 0 & | & -\frac{1}{2} & 0 & 1\end{bmatrix}[/itex]

And this is where I am stumped. Any bump in the right direction will be appreciated.

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# Linear Algebra - Matrix Inverse with an unknown element

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