Math
Explainer
The Exam Shortcut: Check Eigenvalues Fast
Two quick sums can catch an eigenvalue mistake in seconds.
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Lesson notes
Key points
- The trace of a matrix is the sum of its main diagonal entries.
- Eigenvalues, counted with multiplicity, always sum to the trace and multiply to the determinant.
- For rows 2,1 and 1,2, trace four equals one plus three, and determinant three equals one times three.
- A common mistake is a sign error in the polynomial, which these checks expose immediately.
Definitions
- Trace
- tr(A) = sum of diagonal entries a₁₁ + a₂₂ + … + aₙₙ
Takeaway: Eigenvalues sum to the trace and multiply to the determinant.
Quick check: A 2×2 matrix has trace 5 and det 6. Its eigenvalues?
- 1 and 6
- −2 and −3
- 2 and 3
2 + 3 = 5 matches the trace and 2 × 3 = 6 matches the determinant.
TraceSum and productVerifySign errors
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