To express the general solution of the system $Ax = b$ in the form $x = x_r + cx_n$, where $x_r$ is the row space solution and $x_n$ is a basis for the null space of $A$, we proceed with the following steps.

1. Matrix and Vector Definition

Given:

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

2. Check Consistency of the System $Ax = b$

First, let's verify if the system $Ax = b$ is consistent. This involves checking if the vector $b$ lies in the column space of $A$.

$\text{Rank}(A) = 1 \quad \text{and} \quad \text{Rank}([A|b]) = 1$

Since the ranks are equal, the system is consistent, and $b$ lies in the column space of $A$.

3. Row Reduction to Row Echelon Form (REF)

Let's reduce the augmented matrix $[A|b]$ to its row echelon form (REF):

$\begin{pmatrix} -3 & -6 & -6 \\ -9 & -18 & -18 \end{pmatrix} \sim \begin{pmatrix} 1 & 2 & 2 \\ 0 & 0 & 0 \end{pmatrix}$

4. Particular Solution $x_r$ (Row Space Solution)

From the row-reduced form, we obtain the following equation:

$x_1 + 2x_2 = 2$

We can express the particular solution $x_r$ as:

$x_r = \begin{pmatrix} 2 \\ 0 \end{pmatrix} \quad \text{(set } x_2 = 0 \text{)}$

5. 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 the equation:

$x_1 + 2x_2 = 0 \quad \text{or} \quad x_1 = -2x_2$

Thus, the null space solution $x_n$ can be written as:

$x_n = \begin{pmatrix} -2 \\ 1 \end{pmatrix}$

6. General Solution

The general solution to $Ax = b$ is given by:

$x = x_r + c x_n$

Substituting the values of $x_r$ and $x_n$:

$x = \begin{pmatrix} 2 \\ 0 \end{pmatrix} + c \begin{pmatrix} -2 \\ 1 \end{pmatrix}$

Where $c$ is any scalar.

Final Answer

The general solution is:

$x = \begin{pmatrix} 2 \\ 0 \end{pmatrix} + c \begin{pmatrix} -2 \\ 1 \end{pmatrix}$

Would you like any more details or have any questions?

Here are 5 related questions to consider:

1. How do you find the null space of a matrix?
2. What is the significance of the row space and null space in linear algebra?
3. How can you verify if a solution is indeed a solution to a given system?
4. What happens when $b$ does not lie in the column space of $A$?
5. Can a matrix have a null space with more than one dimension?

Tip: Always check the consistency of a system $Ax = b$ by comparing the ranks of $A$ and $[A|b]$ before proceeding with solutions.