Question 31506: Number of Distinct Real Eigenvalues of a 2×2 Matrix

Original question and givens

Over R, the 2×2 matrix is M=[[2, 1], [0, 2]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

• G001 — Over R, the 2×2 matrix is M=[[2, 1], [0, 2]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-eigenvalue

Step-by-step derivation

source roles = [qf-eigenvalue-real-count, reals, real_eigenvalue_count, no-form-field]

derived = [[distinct_real_count_STRING]]

Eigen count ORIGINAL diagonal trace := sum(2, 2)

Eigen count ORIGINAL diagonal trace = 4

Eigen count ORIGINAL ad := mul(2, 2)

Eigen count ORIGINAL ad = 4

Eigen count ORIGINAL bc := mul(1, 0)

Eigen count ORIGINAL bc = 0

Eigen count ORIGINAL ad minus bc := sub(4, 0)

Eigen count ORIGINAL ad minus bc = 4

Eigen count ORIGINAL negative trace := sub(0, 4)

Eigen count ORIGINAL negative trace = -4

Eigen count ORIGINAL trace squared := mul(4, 4)

Eigen count ORIGINAL trace squared = 16

Eigen count ORIGINAL four times determinant := mul(4, 4)

Eigen count ORIGINAL four times determinant = 16

Eigen count ORIGINAL characteristic discriminant := sub(16, 16)

Eigen count ORIGINAL characteristic discriminant = 0

source roles = [[[2, 1], [0, 2]]]

derived = [[1, -4, 4], 0]

source roles = [0]

derived = [1]

Reverse checks

• S011 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
• S010 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
• S009 · premise_and_result_audit — Recompute Eigen count ORIGINAL characteristic discriminant with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Eigen count ORIGINAL four times determinant with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Eigen count ORIGINAL trace squared with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Eigen count ORIGINAL negative trace with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Eigen count ORIGINAL ad minus bc with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Eigen count ORIGINAL bc with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Eigen count ORIGINAL ad with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Eigen count ORIGINAL diagonal trace with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.

Result

1

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.