Question

Let \[ A= \begin{pmatrix} 1 & -2 & -9 & 5 \\ 0 & 1 & 2 & -6 \end{pmatrix}. \] Then, the solution set of the system of equations \[ Ax=0 \] is:

• A. has dimension 2
• B. is a singleton set
• C. is one dimensional
• D. is 4 dimensional

Hint:

The dimension of the solution space of $Ax=0$ represents the number of free variables in the system, which is calculated as: $\text{number of variables} - \text{number of independent equations}$.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
For a homogeneous system of linear equations $Ax = 0$, the dimension of the solution space is the nullity of the matrix $A$.
Key Formula or Approach:
The Rank-Nullity Theorem states:
\[ \text{Rank}(A) + \text{Nullity}(A) = n \]
where $n$ is the number of columns (variables) of the matrix $A$.

Step 2: Detailed Explanation:
Let us analyze the dimensions of the matrix $A$:
$A$ is a $2 \times 4$ matrix, so the number of variables (columns) is $n = 4$.
The rows of $A$ are:
\[ R_1 = (1, -2, -9, 5) \]
\[ R_2 = (0, 1, 2, -6) \]
These two rows are linearly independent because $R_2$ has a leading entry of $1$ at the second column where $R_1$ has $-2$, and $R_2$ has $0$ at the first column where $R_1$ has $1$.
Since the rows are linearly independent, the rank of $A$ is:
\[ \text{Rank}(A) = 2 \]
Now, apply the Rank-Nullity Theorem:
\[ \text{Rank}(A) + \text{Nullity}(A) = 4 \]
\[ 2 + \text{Nullity}(A) = 4 \implies \text{Nullity}(A) = 2 \]
The dimension of the solution space (null space) of $Ax = 0$ is equal to the nullity of $A$.
Therefore, the solution space has dimension 2.

Step 3: Final Answer
The correct option is (A).