Problem 1
The second equation is redundant. Let y=s,z=t; x=3−2s+t, giving infinitely many solutions.
Faculty use
Consistency and Parametric Solution Sets
Reward correct mathematics, explicit dimensions/hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
A row [0 … 0 | c] with c≠0.
Problem 3
Definitions should identify the vector/matrix/space/transformation, dimensions or domain/codomain, and the hypotheses behind classifying unique, infinite, and inconsistent systems. Use the formulas rank(A)=rank([A|b]) for consistency; free variables generate parameters.
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 forces, mixtures, resource balance, geometry, and data representation, 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.