Project ForgeEngineering Mathematics
Linear Algebra
GATE 2019
Hard
Consider the following matrix: R = [[1,2,4,8],[1,3,9,27],[1,4,16,64],[1,5,25,125]]. The absolute value of the product of Eigen values of R is ______.
12
Explanation
The product of a matrix's eigenvalues always equals its determinant. R is a Vandermonde-style matrix whose row i is (1, xᵢ, xᵢ², xᵢ³) for x=2,3,4,5, so det(R) = ∏(xⱼ−xᵢ) over all i<j = (3−2)(4−2)(5−2)(4−3)(5−3)(5−4) = 1×2×3×1×2×1 = 12. So the absolute value of the product of eigenvalues is 12.
Source: GATE CS 2019 (Set-2) (2019)
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