Question

Given:
\[ \begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & 2 \end{bmatrix} \begin{Bmatrix} x_1 \\ x_2 \\ x_3 \end{Bmatrix} = \begin{Bmatrix} 0 \\ 0 \end{Bmatrix} \]
The above system of equations represents a

• A. plane
• B. line
• C. volume
• D. point

Hint:

Find the rank of the coefficient matrix; the solution set has dimension (number of unknowns minus rank).

Answer 1

The Correct Option is B

Step 1: Understand what each equation represents geometrically.
We are given
\[ \begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & 2 \end{bmatrix} \begin{Bmatrix} x_1 \\ x_2 \\ x_3 \end{Bmatrix} = \begin{Bmatrix} 0 \\ 0 \end{Bmatrix} \]
which expands to two linear equations in three unknowns: \[ x_1+x_2+x_3=0 \] \[ x_1+2x_3=0 \] In 3-dimensional space \((x_1,x_2,x_3)\), a single linear equation in three unknowns describes a plane through the origin (the equations are homogeneous, so the right side is zero). Two such planes together describe their line of intersection, unless the planes are identical or parallel.

Step 2: Check the rank of the coefficient matrix.
The coefficient matrix is \(\begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & 2 \end{bmatrix}\). Its two rows, \((1,1,1)\) and \((1,0,2)\), are not scalar multiples of each other, so they are linearly independent. The rank of this \(2\times3\) matrix is therefore \(2\), the maximum possible for two rows.

Step 3: Use rank to find the dimension of the solution set.
For a homogeneous system \(Ax=0\) with \(n\) unknowns and coefficient matrix of rank \(r\), the solution set (null space) has dimension \(n-r\). Here \(n=3\) and \(r=2\), so the dimension is \[ n-r = 3-2 = 1 \] A 1-dimensional solution set through the origin is a line.

Step 4: Confirm by solving explicitly.
From the second equation, \(x_1=-2x_3\). Substituting into the first equation: \(-2x_3+x_2+x_3=0 \implies x_2=x_3\). So every solution has the form \((x_1,x_2,x_3)=(-2t,t,t)\) for a free parameter \(t\), which is exactly the parametric equation of a line through the origin with direction vector \((-2,1,1)\), confirming the rank argument.

Final Answer:
The system represents a line, so option (B) is correct. \[ \boxed{\text{Line}} \]