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(D) Given the matrix equation $AX = B$:$\begin{bmatrix} 1 & -1 & 1 \ 2 & -1 & 0 \ 3 & 3 & -4 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bm

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$3$

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(D) Given the matrix equation $AX = B$:$\begin{bmatrix} 1 & -1 & 1 \ 2 & -1 & 0 \ 3 & 3 & -4 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} = \begin{bmatrix} 1 \ 1 \ 2 \end{bmatrix}$Applying row operations:$R_2 ightarrow R_2 - 2R_1$ and $R_3 ightarrow R_3 - 3R_1$:$\begin{bmatrix} 1 & -1 & 1 \ 0 & 1 & -2 \ 0 & 6 & -7 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} = \begin{bmatrix} 1 \ -1 \ -1 \end{bmatrix}$Now,$R_3 ightarrow R_3 - 6R_2$:$\begin{bmatrix} 1 & -1 & 1 \ 0 & 1 & -2 \ 0 & 0 & 5 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} = \begin{bmatrix} 1 \ -1 \ 5 \end{bmatrix}$From the resulting system of equations:$5x_3 = 5 \implies x_3 = 1$$x_2 - 2x_3 = -1 \implies x_2 - 2(1) = -1 \implies x_2 = 1$$x_1 - x_2 + x_3 = 1 \implies x_1 - 1 + 1 = 1 \implies x_1 = 1$Therefore,$x_1 + x_2 + x_3 = 1 + 1 + 1 = 3$.

If $A = \begin{bmatrix} 1 & -1 & 1 \\ 2 & -1 & 0 \\ 3 & 3 & -4 \end{bmatrix}$,$B = \begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix}$ and $X = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$ such that $AX = B$,then the value of $x_1 + x_2 + x_3$ is:

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