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.