Group the statements below into two sets of equivalent statements. That is, given any specific n × n matrix A (where n ≥ 2), all the statements in one group will be true and all the statements in the other group will be false. (Which group is true and which is false depends of course on A). 1. There exists an n×n matrix B such that AB = I = BA. 2. The homogeneous system Ax = 0 has infinitely many solutions. 3. There exist elementary matrices E1, E2, . . . , Ek such that E1E2 · · · EkA = 8I. 4. For every vector b ∈ R n, the linear system Ax = b has a unique solution. 5. There exist distinct vectors u, v ∈ R n such that Au = 0 and Av = 0. 6. The rank of A is less than n. 7. The reduced row echelon form of A is the identity matrix. 8. det(A) = 0 Write each set as numbers separated by commas, in increasing order. Frist group: Second group:

To group these statements into two sets of equivalent statements, we will analyze each statement and determine whether it describes a situation where the matrix \( A \) is invertible (nonsingular) or not (singular).

### Analysis of Each Statement

1. **Statement 1**: "There exists an \( n \times n \) matrix \( B \) such that \( AB = I = BA \)."  
   This is the definition of an invertible matrix. So, if \( A \) is invertible, this statement is true.

2. **Statement 2**: "The homogeneous system \( Ax = 0 \) has infinitely many solutions."  
   If \( A \) is singular (not invertible), then the homogeneous system has infinitely many solutions. Thus, this statement is true if \( A \) is **not** invertible.

3. **Statement 3**: "There exist elementary matrices \( E_1, E_2, \ldots, E_k \) such that \( E_1E_2 \cdots E_kA = 8I \)."  
   For this statement to be true, \( A \) must be invertible because elementary matrices correspond to row operations that, when applied to an invertible \( A \), yield a nonzero scalar multiple of the identity matrix (here, \( 8I \)).

4. **Statement 4**: "For every vector \( b \in \mathbb{R}^n \), the linear system \( Ax = b \) has a unique solution."  
   This is true if and only if \( A \) is invertible, as invertibility ensures a unique solution for every \( b \).

5. **Statement 5**: "There exist distinct vectors \( u, v \in \mathbb{R}^n \) such that \( Au = 0 \) and \( Av = 0 \)."  
   If \( A \) is singular, then there exists a nontrivial null space, meaning there are distinct vectors that satisfy \( A u = 0 \). Therefore, this statement is true if \( A \) is **not** invertible.

6. **Statement 6**: "The rank of \( A \) is less than \( n \)."  
   If \( A \) is singular, it does not have full rank, so the rank of \( A \) would be less than \( n \). This statement is true if \( A \) is **not** invertible.

7. **Statement 7**: "The reduced row echelon form of \( A \) is the identity matrix."  
   If \( A \) is invertible, then it can be row reduced to the identity matrix. Thus, this statement is true if \( A \) is invertible.

8. **Statement 8**: "det(A) = 0"  
   A matrix is singular if and only if its determinant is zero. Therefore, this statement is true if \( A \) is **not** invertible.

### Grouping the Statements

Based on this analysis, we can group the statements as follows:

- **Group 1** (Statements true if \( A \) is invertible): **1, 3, 4, 7**
- **Group 2** (Statements true if \( A \) is not invertible): **2, 5, 6, 8**

### Final Answer

- **First group**: 1, 3, 4, 7
- **Second group**: 2, 5, 6, 8

Would you like further clarification on any part of this grouping? 

Here are 5 related questions:
1. How does the invertibility of a matrix affect the number of solutions to a linear system?
2. Why is the determinant used as a criterion for matrix invertibility?
3. Can a matrix with a nonzero determinant ever have a nontrivial null space?
4. How are elementary matrices related to invertibility and row operations?
5. What are the implications of a matrix’s rank on the solutions of its associated linear system?

**Tip**: The determinant of a matrix provides insight into whether the matrix is invertible. If the determinant is zero, the matrix is singular (non-invertible), indicating a lack of unique solutions in its linear systems.

Group the statements below into two sets of equivalent statements. That is, given any specific n × n matrix A (where n ≥ 2), all the statements in one group will be true and all the statements in the other group will be false. (Which group is true and which is false depends of course on A). 1. There exists an n×n matrix B such that AB = I = BA. 2. The homogeneous system Ax = 0 has infinitely many solutions. 3. There exist elementary matrices E1, E2, . . . , Ek such that E1E2 · · · EkA = 8I. 4. For every vector b ∈ R n, the linear system Ax = b has a unique solution. 5. There exist distinct vectors u, v ∈ R n such that Au = 0 and Av = 0. 6. The rank of A is less than n. 7. The reduced row echelon form of A is the identity matrix. 8. det(A) = 0 Write each set as numbers separated by commas, in increasing order. First group: Second group:

Analyze and group statements about an n × n matrix A based on matrix invertibility. Covers key linear algebra concepts, including determinants, the rank-nullity theorem, and invertible matrix properties, making it ideal for college-level studies.

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