The product of the non-zero eigenvalues of the matrix is ____ [1 0 0 0 1] [0 1 1 1 0] [0 1 1 1 0] [0 1 1 1 0] [1 0 0 0 1]

GATE 2014 · Engineering Mathematics · Eigen Value · medium

Answer: The product of the non-zero eigenvalues is 6.

1. Identify rank by spotting dependent rows: Row1 = [1,0,0,0,1], Row5 = [1,0,0,0,1] => Row1 = Row5 (dependent). Rows 2, 3, 4 are all [0,1,1,1,0] => identical. So independent rows: r1 = [1,0,0,0,1] and r2 = [0,1,1,1,0]. Rank = 2.
2. Compute effective 2x2 matrix (Gram matrix of row vectors): r1 . r1 = 1+0+0+0+1 = 2; r2 . r2 = 0+1+1+1+0 = 3; r1 . r2 = 0. G = [[2,0],[0,3]]. But M = R^T * R (5x5), and non-zero eigenvalues of M equal those of R*R^T (by rank-factorization). G is diagonal [[2,0],[0,3]] so eigenvalues are 2 and 3.
3. Compute product of non-zero eigenvalues: Product = 2 * 3 = 6.