A=pmatrix 1&1&0\\0& 2&1\\0&0& 3pmatrix with real λ i. Which hold?

For every A some real α,β satisfy A²+αA+βI=0

GATE 2026 ST – Question 50

Matrix Theory · Eigenvalues and Matrix Decompositions · 2 marks · Multiple select

$A=\begin{pmatrix}\lambda_1&1&0\\0&\lambda_2&1\\0&0&\lambda_3\end{pmatrix}$ with real λ_i. Which hold?

Never diagonalizable
Diagonalizable exactly when λ₁,λ₂,λ₃ are distinct
Each eigenspace at a λ_i has dimension 1
For every A some real α,β satisfy A²+αA+βI=0

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Correct answer: (B) Diagonalizable exactly when λ₁,λ₂,λ₃ are distinct; (C) Each eigenspace at a λ_i has dimension 1

Explanation

Solving (A−λI)v=0 from the bottom row upward gives nullity 1 for each eigenvalue, even when values repeat. Thus diagonalizability requires three distinct eigenvalues, and distinctness is sufficient. Alternatively e₃,Ae₃,A²e₃ are independent, so A is cyclic with minimal polynomial of degree 3, refuting D. **B,C**.

GATE 2026 ST – Question 50

Explanation

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