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 _______

GATE 2019 · Engineering Mathematics · Eigen Value · medium

Answer: 12

1. Identify matrix structure: R rows: x_1=2, x_2=3, x_3=4, x_4=5. Each row: [1, x, x^2, x^3]. This is a Vandermonde matrix.
2. Apply Vandermonde determinant formula: Pairs (i<j): (1,2)=3-2=1, (1,3)=4-2=2, (1,4)=5-2=3, (2,3)=4-3=1, (2,4)=5-3=2, (3,4)=5-4=1. det(R) = 1*2*3*1*2*1 = 12
3. Product of eigenvalues = det(R): |product of eigenvalues| = |det(R)| = |12| = 12