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MMLU-Pro / 8812 / Let M be a 5 x 5 real matrix. Exactly four of the following five conditions on M…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

question

Let M be a 5 x 5 real matrix. Exactly four of the following five conditions on M are equivalent to each other. Which of the five conditions is equivalent to NONE of the other four?

category

math

options

  1. The column vectors of M form a basis for R^5.
  2. For any two distinct column vectors u and v of M, the set {u, v} is linearly independent.
  3. The trace of M is nonzero.
  4. The eigenvalues of M are all nonzero.
  5. The homogeneous system Mx = 0 has only the trivial solution.
  6. The determinant of M is nonzero.
  7. The inverse of M exists.
  8. The transpose of M is equal to M.
  9. The rank of M is 5.
  10. The system of equations Mx = b has a unique solution for each real 5 x 1 column vector b.

src

ori_mmlu-college_mathematics

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Source and history

upstream_id / Original problem ID
8812
task_id
c974216c-83b6-5389-b5af-a441978a5a8f
task_key
default--test--8812
task_revision_id
1
config / split
default / test

Official source

initial import