GATE 2019 CS – Question 54
Consider the following matrix:
$$R=\begin{bmatrix}1&2&4&8\\1&3&9&27\\1&4&16&64\\1&5&25&125\end{bmatrix}$$
The absolute value of the product of Eigen values of $R$ is ________.
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Correct answer: 12
Explanation
The product of the eigenvalues equals the determinant. Each row of $R$ is $(1,x,x^2,x^3)$ for $x=2,3,4,5$, so $R$ is a Vandermonde matrix and $\det R=\prod_{i<j}(x_j-x_i)=(3-2)(4-2)(5-2)(4-3)(5-3)(5-4)=1\cdot2\cdot3\cdot1\cdot2\cdot1=12$.