The first equation makes −i an eigenvalue. Real entries give the conjugate eigenvalue i. Their product is 1, so the third eigenvalue is 3. The trace of A² is the sum of eigenvalue squares, even without diagonalizability: (−i)²+i²+3²=−1−1+9=**7**.
More Eigenvalues and Canonical Forms questions
GATE 2023 MA Q60– A=(a ij) is real 3×3 with A(1,2,1)ᵀ=2(1,2,1)ᵀ, A(0,1,1)ᵀ=2(0,1,1)ᵀ and…
GATE 2022 MA Q11– M∈ℂ³ˣ³ has characteristic polynomial λ³+αλ²+βλ−1, with α+β≠0. Which identity holds?
GATE 2022 MA Q65– A real 3×3 M satisfies M²=2M+3I and det M=−9. Find trace M.
GATE 2021 MA Q62– V is the real vector space of polynomials p=a₀+a₁x+a₂x². Define…
GATE 2026 MA Q28– T reflects ℝ³ through a two-dimensional subspace. P is its matrix in the standard basis.…
GATE 2026 MA Q48– Nonzero real 3×3 matrices P,Q are simultaneously upper triangularizable by the same…