Let's re-examine the solution to ensure accuracy. The expected form is \( x = \begin{pmatrix} 0.4 \\ 0.8 \end{pmatrix} + c \begin{pmatrix} -2 \\ 1 \end{pmatrix} \).

### Step 1: Given Matrix and Vector
The matrix \( A \) and vector \( b \) are:

\[
A = \begin{pmatrix} -3 & -6 \\ -9 & -18 \end{pmatrix}, \quad b = \begin{pmatrix} -6 \\ -18 \end{pmatrix}
\]

### Step 2: Row Reduction to Row Echelon Form (REF)
We'll perform row reduction on the augmented matrix \( [A|b] \):

\[
\begin{pmatrix} -3 & -6 & -6 \\ -9 & -18 & -18 \end{pmatrix} \sim \text{(Divide the first row by } -3)\)
\[
\begin{pmatrix} 1 & 2 & 2 \\ -9 & -18 & -18 \end{pmatrix}
\]
\[
\text{Now add 9 times the first row to the second row:}
\]
\[
\begin{pmatrix} 1 & 2 & 2 \\ 0 & 0 & 0 \end{pmatrix}
\]

This gives us the system of equations:

\[
x_1 + 2x_2 = 2
\]

### Step 3: Particular Solution \( x_r \)
We can solve for \( x_1 \) in terms of \( x_2 \):

\[
x_1 = 2 - 2x_2
\]

If we set \( x_2 = t \), then:

\[
x_1 = 2 - 2t
\]

Choosing \( t = 0.8 \) as the free variable:

\[
x_1 = 2 - 2(0.8) = 2 - 1.6 = 0.4
\]

So, the particular solution \( x_r \) is:

\[
x_r = \begin{pmatrix} 0.4 \\ 0.8 \end{pmatrix}
\]

### Step 4: Null Space Solution \( x_n \)
To find the null space \( \text{null}(A) \), solve \( Ax = 0 \):

\[
\begin{pmatrix} -3 & -6 \\ -9 & -18 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}
\]

This leads to:

\[
x_1 + 2x_2 = 0 \quad \Rightarrow \quad x_1 = -2x_2
\]

Thus, the null space solution \( x_n \) is:

\[
x_n = \begin{pmatrix} -2 \\ 1 \end{pmatrix}
\]

### Step 5: General Solution
The general solution can now be written as:

\[
x = x_r + c x_n = \begin{pmatrix} 0.4 \\ 0.8 \end{pmatrix} + c \begin{pmatrix} -2 \\ 1 \end{pmatrix}
\]

### Final Verification
This matches the expected solution:

\[
x = \begin{pmatrix} 0.4 \\ 0.8 \end{pmatrix} + c \begin{pmatrix} -2 \\ 1 \end{pmatrix}
\]

Would you like more details on any of these steps or have any further questions?

Here are 5 related questions:

1. How does the choice of free variables affect the particular solution?
2. What does the null space of a matrix represent geometrically?
3. Can every system \( Ax = b \) be solved in the form \( x = x_r + c x_n \)?
4. How does the row space relate to the column space in linear algebra?
5. What are the conditions for a unique solution in a linear system?

**Tip:** Always double-check your calculations and assumptions, especially when interpreting free variables, to ensure the correctness of the solution.

Explore the solution process for the expected form in linear algebra problems. This detailed guide covers matrix operations, vector spaces, and row reduction techniques. Suitable for college-level students and anyone interested in linear algebra concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation