Only from 2nd question onwards and only steps and answer no explanation 

Certainly, here are the steps and answers from the 2nd question onwards:

### Problem 2: Examine the Consistency of the Linear System \(AX = 0\) and Solve It

Given matrix \( A \):
\[
A = \begin{bmatrix} 
1 & 2 & 3 \\
2 & 4 & 6 \\
3 & 6 & 9 
\end{bmatrix}
\]

1. **Row operations to get echelon form** (as done previously):
   \[
   \begin{bmatrix}
   1 & 2 & 3 \\
   0 & 0 & 0 \\
   0 & 0 & 0 
   \end{bmatrix}
   \]

2. **General solution**:
   \[
   \begin{aligned}
   x_1 &= -2t - 3s \\
   x_2 &= t \\
   x_3 &= s 
   \end{aligned}
   \]
   where \( t \) and \( s \) are free parameters.

### Problem 3: Find Rank by Using Normal Form

Given matrix \( A \):
\[
A = \begin{bmatrix} 
1 & 2 & 3 \\
2 & 5 & 6 
\end{bmatrix}
\]

1. **Row operation**:
   \[
   R_2 \rightarrow R_2 - 2R_1
   \]

2. **Normal form**:
   \[
   A = \begin{bmatrix}
   1 & 2 & 3 \\
   0 & 1 & 0 
   \end{bmatrix}
   \]

**Rank** = 2.

Explore how to examine the consistency of a linear system AX = 0 and solve it using row operations to achieve row echelon form. Includes a detailed explanation of the general solution with free parameters. Suitable for advanced undergraduate students studying linear algebra.

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