The characteristic polynomial is λ²−4λ+3=(λ−1)(λ−3). Distinct real eigenvalues give M=S diag(1,3)S⁻¹. Every polynomial p(M)=S diag(p(1),p(3))S⁻¹ is diagonalizable, proving both statements. **D**.
More Eigenvalues and Canonical Forms questions
GATE 2023 MA Q22– T:ℝ⁴→ℝ⁴ has kernel 4x₁+3x₂+2x₃+x₄=0 and rank(T−3I)=3. Find its minimal polynomial.
GATE 2023 MA Q35– Let M=pmatrix3&-1&-2\\0&2&4\\0&0&1pmatrix. If 6M⁻¹=M²−6M+αI, find α.
GATE 2023 MA Q59– A is a real 3×3 matrix with det(A+iI)=0 and det A=3. Find trace(A²).
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.