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GATE 2024 ST – Question 49

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

Which claims hold for real 3×3 matrices?

  1. Three distinct eigenvalues imply diagonalizability.
  2. Diagonalizable A² implies diagonalizable A.
  3. If [[0,a,b],[0,0,c],[0,0,0]] is diagonalizable then a=b=c=0.
  4. Every diagonalizable upper triangular A satisfies AAᵀ=AᵀA.

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Show answer and explanation

Correct answer: (A) Three distinct eigenvalues imply diagonalizability.; (C) If [[0,a,b],[0,0,c],[0,0,0]] is diagonalizable then a=b=c=0.

Explanation

Distinct eigenvalues give independent eigenvectors, proving A over ℂ; real diagonalization additionally requires all eigenvalues real. A nonzero size-two nilpotent block plus a zero block has diagonalizable square zero but is not diagonalizable, refuting B. The matrix in C is nilpotent with only eigenvalue zero; a diagonalizable such matrix is zero. D fails for upper triangular [[1,1,0],[0,2,0],[0,0,3]], which has distinct real eigenvalues but is not normal. **A,C** under the standard complex-eigenvalue convention.

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