Math Problem Statement
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:
Solution
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Invertibility
Determinants
Rank of a Matrix
Systems of Linear Equations
Formulas
Invertibility: AB = I = BA
Homogeneous System Solutions: Ax = 0
Determinant Condition for Invertibility: det(A) ≠ 0
Theorems
Invertible Matrix Theorem
Rank-Nullity Theorem
Suitable Grade Level
College Level - Linear Algebra
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