What are the eigenvalues of the matrix P = [[1,0,0],[a,1,0],[0,a,1]]? A. a_n, a, a (i.e., 1, 1, 1) B. -a, -a, a C. 0, 2a, 2a D. -a, -2a, 2a

GATE 2006 · Engineering Mathematics · Eigen Value · medium

Answer: All three eigenvalues equal 1. Answer: A.

1. Identify matrix type: P = [[1,0,0],[a,1,0],[0,a,1]] is lower triangular (zeros above diagonal)
2. Read off eigenvalues: Diagonal entries: P[1][1]=1, P[2][2]=1, P[3][3]=1 => eigenvalues are 1, 1, 1
3. Verify via characteristic polynomial: det([[1-lambda,0,0],[a,1-lambda,0],[0,a,1-lambda]]) = (1-lambda)^3 = 0 => lambda=1 (triple root)