To solve the system [tex]\( A x = 0 \)[/tex] for the matrix
[tex]\[
A = \left[\begin{array}{rrrr}
2 & -3 & -6 & -4 \\
1 & 5 & -3 & 11 \\
2 & 7 & -6 & 16
\end{array}\right]
\][/tex]
we first need to determine the rank and nullity of the matrix, along with the number of parameters required.
### Step 1: Determine the Rank
The [tex]\(\textbf{rank}\)[/tex] of a matrix is the maximum number of linearly independent rows or columns in the matrix. The rank of the matrix [tex]\( A \)[/tex] is [tex]\( 2 \)[/tex].
### Step 2: Determine the Nullity
The [tex]\(\textbf{nullity}\)[/tex] of a matrix is given by the number of columns minus the rank of the matrix. For our matrix [tex]\( A \)[/tex]:
- Number of columns of [tex]\( A \)[/tex] = [tex]\( 4 \)[/tex]
- Rank of [tex]\( A \)[/tex] = [tex]\( 2 \)[/tex]
Therefore, the nullity is calculated as:
[tex]\[
\text{nullity} = \text{number of columns} - \text{rank} = 4 - 2 = 2
\][/tex]
### Step 3: Introduce Parameters
The [tex]\(\textbf{number of parameters}\)[/tex] corresponds to the number of free variables in the solution to the system [tex]\( A x = 0 \)[/tex], which is equal to the nullity of the matrix. Hence, we introduce [tex]\( 2 \)[/tex] parameters.
### Step 4: Dimension of the Vector Space
The [tex]\(\textbf{dimension of the vector space}\)[/tex] spanned by the columns of the matrix [tex]\( A \)[/tex] is simply the number of columns of the matrix, which is [tex]\( 4 \)[/tex].
### Final Solution:
- [tex]\(\text{Number of parameters you introduce} = 2\)[/tex]
- [tex]\(\text{Nullity} = 2\)[/tex]
- [tex]\(\text{Rank} = 2\)[/tex]
- [tex]\(\text{Dimension of the vector space} = 4\)[/tex]
So, the answers are:
[tex]\[
\boxed{2}, \boxed{2}, \boxed{2}, \boxed{4}
\][/tex]
