Let
A
=
⎡
⎢
⎣
1
2
−
6
2
0
0
3
−
12
0
0
0
0
⎤
⎥
⎦
. What is the solution space of
A
?

The given matrix \( A \) is:

\[
A = \begin{pmatrix}
1 & 2 & -6 \\
2 & 0 & 0 \\
3 & -12 & 0 \\
0 & 0 & 0
\end{pmatrix}
\]

We are tasked with determining the solution space of this matrix, which involves finding the null space of \( A \), i.e., the set of vectors \( \mathbf{x} \) that satisfy:

\[
A \mathbf{x} = 0
\]

### Step 1: Write the system of equations
Let's assume \( \mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} \). We can write the matrix equation \( A \mathbf{x} = 0 \) as the following system of equations:

\[
\begin{aligned}
1x_1 + 2x_2 - 6x_3 &= 0 \\
2x_1 &= 0 \\
3x_1 - 12x_2 &= 0
\end{aligned}
\]

### Step 2: Solve the system

- From the second equation: \( 2x_1 = 0 \), we have \( x_1 = 0 \).
- Substituting \( x_1 = 0 \) into the first equation: \( 0 + 2x_2 - 6x_3 = 0 \), which simplifies to:
  \[
  2x_2 - 6x_3 = 0 \implies x_2 = 3x_3
  \]
- The third equation becomes \( 3(0) - 12x_2 = 0 \), which is automatically satisfied.

### Step 3: Solution space

Thus, the solution space (null space) is given by:

\[
\mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} 0 \\ 3x_3 \\ x_3 \end{pmatrix} = x_3 \begin{pmatrix} 0 \\ 3 \\ 1 \end{pmatrix}
\]

The null space is spanned by the vector \( \begin{pmatrix} 0 \\ 3 \\ 1 \end{pmatrix} \).

### Conclusion

The solution space of the matrix \( A \) is the set of all scalar multiples of the vector \( \begin{pmatrix} 0 \\ 3 \\ 1 \end{pmatrix} \). In other words, the null space of \( A \) is:

\[
\text{Null}(A) = \text{span} \left( \begin{pmatrix} 0 \\ 3 \\ 1 \end{pmatrix} \right)
\]

Would you like more details or have any questions?

### Related questions:
1. What is the rank of matrix \( A \)?
2. How does one determine the nullity of a matrix?
3. Can a non-square matrix have a null space? If so, how is it found?
4. What are the implications of a matrix having a non-trivial null space?
5. How does the null space relate to the kernel of a linear transformation?

### Tip:
The dimension of the null space of a matrix is called the **nullity**, and it provides insight into the "degree of freedom" in the system of linear equations represented by the matrix.

Let A = \begin{pmatrix} 1 & 2 & -6 \\ 2 & 0 & 0 \\ 3 & -12 & 0 \\ 0 & 0 & 0 \end{pmatrix}. What is the solution space of A?

Explore how to determine the solution space of the matrix A using its null space. This linear algebra problem covers concepts like vector spaces and the null space theorem, with step-by-step solutions. Ideal for college-level students.

Math Problem Statement

Let A

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation