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2026 · Exam 94Difficulty 4In progress

Diagonalizability over different fields

Let A be a 3 × 3 matrix satisfying A² = 3A⁻¹. Determine every possible value of det(A), and explain whether your conclusion changes over ℝ and ℂ.

  1. 1. Begin by multiplying both sides by A.
  2. 2. Use the determinant identities det(A³) = det(A)³ and det(3I) = 3³.

I can see the approach you wrote on iPad. You have the right first step: A³ = 3I.

Your determinant chain is correct:

  • A² = 3A⁻¹
  • A³ = 3I ✓
  • det(A)³ = det(3I) = 27 ✓

Before solving the cube root, what restriction does det(A) ∈ ℝ place on the possible values?

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Recursion
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Linear algebra knowledge test1 of 3
If A³ = 3I for a 3 × 3 matrix A, which determinant equation follows?

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