Problem 1
It is singular, has dependent columns, nontrivial nullspace, and zero is an eigenvalue.
Faculty use
Determinant, Invertibility & Eigenvalue Zero
Reward correct mathematics, explicit dimensions/hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
No; A is invertible, so every b has exactly one solution.
Problem 3
Definitions should identify the vector/matrix/space/transformation, dimensions or domain/codomain, and the hypotheses behind connecting zero determinant to singular structure. Use the formulas detA≠0 ⇔ A invertible; detA=product of eigenvalues.
Problem 4
A strong error analysis identifies the first step that violates a definition, dimension, theorem hypothesis, or matrix-operation rule; then repairs that step and re-verifies the final result.
Problem 5
Application responses vary. Require defined variables, a compatible matrix/vector model connected to area and volume scaling, coordinate change, invertibility tests, and geometric design, a complete computation, interpretation, and limitations.
Problem 6
Accept two independent checks appropriate to the task: reconstruction, residual, RREF comparison, determinant/rank test, orthogonality, substitution, eigenpair check Av=λv, factorization product, or a second method.
Scoring guidance
Six problems × 10 points: 5 mathematical accuracy, 2 structure/justification, 2 verification/interpretation, 1 notation and communication.