The correct option is line.

Step-by-step Explanation:

From the provided image, we are given a homogeneous system of linear equations in matrix form:

$$\left[\begin{array}{ccc}1& 1& 1\\ 1& 0& 2\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]$$

This matrix equation translates to a system of two linear equations with three variables (_x1, _x2, and _x3):

1) $x_1+x_2+x_3=0$

2) $x_1+2x_3=0$

Geometrically, each of these linear equations represents a plane in three-dimensional space (${ℝ}^3$) passing through the origin (0, 0, 0).

The normal vectors to these two planes are:

${\mathbf{n}}_1=(1,1,1)$

${\mathbf{n}}_2=(1,0,2)$

Since the normal vectors ${\mathbf{n}}_1$ and ${\mathbf{n}}_2$ are not scalar multiples of each other, the two planes are not parallel. In three-dimensional space, the intersection of two non-parallel planes is a line.

Alternatively, we can analyze the solution using the Rank-Nullity Theorem. The coefficient matrix A is of size 2 × 3:

$$A=\left[\begin{array}{ccc}1& 1& 1\\ 1& 0& 2\end{array}\right]$$

Since the two rows of matrix A are linearly independent, the rank of the matrix is 2. The number of variables (columns) is n = 3.

According to the Rank-Nullity Theorem:

$\text{Nullity}\left(A\right)=n-\text{Rank}\left(A\right)=3-2=1$

The nullity represents the dimension of the solution space. Since the dimension of the solution space is 1, the system of equations represents a one-dimensional geometric object, which is a line passing through the origin.