Question:

Suppose that the linear system $$\begin{bmatrix} 2 & 0 & -1 \\ 0 & c & 1 \\ 1 & 3 & -2 \end{bmatrix} \vec{x}=\vec{b}= \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$$ is inconsistent for some vector $\vec{b}$. Determine c.

$$\color{crimson}{------------------------------------------}$$

What I did:

I put this into RREF and got :

$$\begin{bmatrix} 2 & 0 & -1 & |x_1\\ 0 & c & 1 & | x_2 \\ 0 & 2+c & 0 & |2x_3-x_1+x_2 \end{bmatrix}$$

I do not know what to do next.

Question:

Suppose that the linear system $$\begin{bmatrix} 2 & 0 & -1 \\ 0 & c & 1 \\ 1 & 3 & -2 \end{bmatrix} \vec{x}=\vec{b}= \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$$ is inconsistent for some vector $\vec{b}$. Determine c.

$$\color{crimson}{------------------------------------------}$$

What I did:

I put this into RREF and got :

$$\left[\begin{array}{rrr|r} 2 & 0 & -1 & x_1\\ 0 & c & 1 & x_2 \\ 0 & 2+c & 0 & 2x_3-x_1+x_2 \end{array}\right]$$

I do not know what to do next.

Question:

Suppose that the linear system $$\begin{bmatrix} 2 & 0 & -1 \\ 0 & c & 1 \\ 1 & 3 & -2 \end{bmatrix} \vec{x}=\vec{b}= \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$$ is inconsistent for some vector $\vec{b}$. Determine c.

$$\color{crimson}{------------------------------------------}$$

What I did:

I put this into RREF and got :

$$\begin{bmatrix} 2 & 0 & -1 & |x_1\\ 0 & c & 1 & | x_2 \\ 0 & 2+c & 0 & |2x_3-x_1+x_2 \end{bmatrix}$$

I do not know what to do next.

Question:

Suppose that the linear system $$\begin{bmatrix} 2 & 0 & -1 \\ 0 & c & 1 \\ 1 & 3 & -2 \end{bmatrix} \vec{x}=\vec{b}= \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$$ is inconsistent for some vector $\vec{b}$. Determine c.

$$\color{crimson}{------------------------------------------}$$

What I did:

I put this into RREF and got :

$$\begin{bmatrix} 2 & 0 & -1 & |x_1\\ 0 & c & 1 & | x_2 \\ 0 & 2+c & 0 & |2x_3-x_1+x_2 \end{bmatrix}$$

I do not know what to do next.