The approximate eigenvalue of the matrix
\[ A = \begin{bmatrix} -15 & 4 & 3 \\ 10 & -12 & 6 \\ 20 & -4 & 2 \end{bmatrix} \]
obtained after two iterations of Power method, with the initial vector \([1\ 1\ 1]^{T}\), is
GATE Mathematics - 2012,
Answer: C
After two iterations the estimate is 10.548.
Each step multiplies by \(A\) and scales by the entry of largest magnitude.
\[ \begin{aligned} A[1, 1, 1]^{T} &= [-8, 4, 18]^{T} \\ &= 18\,[-0.444, 0.222, 1]^{T} \end{aligned} \]
\[ \begin{aligned} A[-0.444, 0.222, 1]^{T} \\ = [10.548, -1.104, -7.768]^{T} \end{aligned} \]
The first component is \(6.660 + 0.888 + 3 = 10.548\), the entry of largest magnitude, so the eigenvalue estimate after two iterations is 10.548 (with the vector rounded to three decimals, as the options assume). 7.768 is the magnitude of the third component, not the largest one, and 18 from the first iteration is not offered; 9.468 and 19.468 do not arise from the iteration.
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