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GATE 2020 MA – Question 49

Numerical Analysis · Linear Systems · 2 marks · Multiple choice

Let $$M=\begin{pmatrix}\alpha&3&0\\\beta&3&1\\0&1&2\end{pmatrix}.$$ I: there is a real lower-triangular L with M = LLᵀ. II: Gauss–Seidel iteration for Mx = b converges for every initial vector. Which statement is correct?

  1. I is false when α > 9/2, β = 3.
  2. II is false when α > 9/2, β = −1.
  3. II is false when α = 4, β = 3/2.
  4. I is true when α = 5, β = 3.

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Correct answer: (D) I is true when α = 5, β = 3.

Explanation

For α = 5, β = 3, M is symmetric and its leading principal minors are 5, $15-9=6$, and $5(5)-18=7$, all positive. It is positive definite and has a Cholesky factorization LLᵀ, proving D. The same test disproves A for α > 9/2, β = 3. The Gauss–Seidel iteration has spectral radius $|\beta/\alpha+1/6|$, which is below 1 for the cases in B and C, so their asserted failures are false.

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