Math Problem Statement

Solution

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

A=[2672139].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 x\mathbf{x} such that when multiplied by the matrix AA, it results in the zero vector, i.e.,

Ax=0.A \mathbf{x} = \mathbf{0}.

Since the matrix AA 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 Ax=0A \mathbf{x} = 0 is x=[0.94870.3162]\mathbf{x} = \begin{bmatrix} 0.9487 \\ -0.3162 \end{bmatrix}.

This vector represents a direction in the null space of matrix AA. Any scalar multiple of this vector will also satisfy the equation Ax=0A \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 Ax=0A \mathbf{x} = 0?
  3. How is the rank of matrix AA 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 Ax=0A \mathbf{x} = 0. For a matrix with dependent rows or columns, always expect nontrivial solutions.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Operations
Null Space

Formulas

Matrix-vector multiplication

Theorems

Rank-nullity theorem

Suitable Grade Level

Advanced Undergraduate