Consider the following matrix: \(R = \begin{bmatrix} 1 & 2 & 4 & 8 \\ 1 & 3 &…

Key idea: the product of the eigenvalues equals the determinant of the matrix.

This matrix is a Vandermonde matrix built from the nodes 2, 3, 4, 5 (each row is [1, x, x^2, x^3]). The determinant of a Vandermonde matrix with nodes x1, x2, x3, x4 is the product over all pairs (xi, xj) with i<j of (xj - xi).

• Compute the pairwise differences for nodes 2, 3, 4, 5: (2,3)=1, (2,4)=2, (2,5)=3, (3,4)=1, (3,5)=2, (4,5)=1

• Multiply these differences: 1 × 2 × 3 × 1 × 2 × 1 = 12

Therefore the determinant equals 12, so the absolute value of the product of the eigenvalues is 12.