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The problem asks to find a nontrivial solution of the equation \( A \mathbf{x} = 0 \), where

\[
A = \begin{bmatrix} -2 & -6 \\ 7 & 21 \\ -3 & -9 \end{bmatrix}.
\]

The hint suggests thinking of this equation as a vector equation. 

### Approach:
To find a nontrivial solution, we need to determine a vector \( \mathbf{x} \) such that when multiplied by the matrix \( A \), it results in the zero vector, i.e.,

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

Since the matrix \( A \) has dependent rows (as indicated by the fact that each row is a multiple of another), it will have infinitely many solutions, one of which is nontrivial. We can use inspection to find a nontrivial solution.

By inspection, let's check for a solution.It appears that my Python environment was reset, causing my previous variables to be lost. Let me redefine everything and find the nontrivial solution again.A nontrivial solution to the equation \( A \mathbf{x} = 0 \) is \( \mathbf{x} = \begin{bmatrix} 0.9487 \\ -0.3162 \end{bmatrix} \).

This vector represents a direction in the null space of matrix \( A \). Any scalar multiple of this vector will also satisfy the equation \( A \mathbf{x} = 0 \), providing other nontrivial solutions.

### Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. What does the null space of a matrix represent?
2. How can you determine the number of solutions to \( A \mathbf{x} = 0 \)?
3. How is the rank of matrix \( A \) related to the dimension of its null space?
4. What is the significance of finding a nontrivial solution to a homogeneous system of linear equations?
5. How do row operations affect the null space of a matrix?

**Tip:** The null space gives all solutions to \( A \mathbf{x} = 0 \). For a matrix with dependent rows or columns, always expect nontrivial solutions.

Explore how to find nontrivial solutions to matrix equations using a detailed example problem. This problem covers linear algebra concepts such as matrix-vector multiplication and the rank-nullity theorem. Suitable for advanced undergraduate students.

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