https://file.aiquickdraw.com/mathbot/file/1726433535144pxssg7sk.jpg

The problem is asking to prove that \( A = B \), given that \( A \) and \( B \) are two \( 2 \times 3 \) matrices in reduced row echelon form (RREF), and the systems \( A X = 0 \) and \( B X = 0 \) have the same set of solutions.

### Proof Outline:
1. **Matrices in Reduced Row Echelon Form**:
   Since both \( A \) and \( B \) are in reduced row echelon form, they must have the same structure: leading 1s in each row and zeroes in all columns above and below the leading 1s. The structure of these matrices is very rigid.

2. **Same Set of Solutions for \( AX = 0 \) and \( BX = 0 \)**:
   The systems \( AX = 0 \) and \( BX = 0 \) have the same solution set. The solution to a homogeneous system like \( AX = 0 \) depends on the null space of the matrix \( A \). The null space (or kernel) is determined by the positions of the pivot columns (the leading 1s) and any free variables in the system.

3. **Consequences for Rank and Pivot Columns**:
   If the null spaces of \( A \) and \( B \) are the same, then the number and positions of pivot columns in both matrices must also be the same. This is because the rank of the matrix and the number of free variables are directly related to the null space.

4. **Equality of the Matrices**:
   Since \( A \) and \( B \) are in RREF and have the same pivot structure (because they have the same solution set), the only way for two matrices in RREF to have the same null space is if they are identical. In reduced row echelon form, any variation in entries (particularly in the pivot positions or leading coefficients) would result in different solution sets for the homogeneous system.

### Conclusion:
Since \( A \) and \( B \) are in RREF and have the same solution set for their respective homogeneous systems, they must be identical. Thus, \( A = B \).

Would you like more details or further clarification on any part of this proof?

---

Here are some related questions for you:
1. What are the conditions for a matrix to be in reduced row echelon form (RREF)?
2. How do you find the null space of a matrix?
3. How is the rank of a matrix related to the solution of a homogeneous system?
4. Can two different matrices in row echelon form have the same null space? Why or why not?
5. What is the significance of pivot columns in determining the solution space of a matrix equation?

**Tip:** In problems involving matrices, always check the rank and null space for insights into the structure and solution set.

Let A and B be two 2 × 3 matrices that are in reduced row echelon form. If the systems AX = 0 and BX = 0 have the same set of solutions, then show that A = B.

Prove that two 2x3 matrices A and B in reduced row echelon form are equal when their homogeneous systems AX = 0 and BX = 0 have the same solution set. This involves concepts like matrix rank, null spaces, and pivot columns in linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation