Which one of the following statements is NOT true for a square matrix A? A. If A is real symmetric, the eigen values of A are the diagonal elements of it. B. If all the principal minors of A are positive, all the eigen values of A are also positive. C. If A is real symmetric, then there exists an orthogonal matrix Q such that A = Q*D*Q^T where D is a diagonal matrix. D. If A is real, then the eigen values of A and A^T are the same.

GATE 2014 · Engineering Mathematics · Eigen Value · medium

Answer: Statement A is NOT true. Answer: A.

1. Test Statement A with counterexample: A = [[0,1],[1,0]]: symmetric, diagonal entries = 0,0. Eigenvalues: lambda^2 - 0 - 1 = 0 => lambda = +/-1. Diagonal entries (0,0) != eigenvalues (+/-1). Statement A is FALSE.
2. Verify Statement D is TRUE: det(A - lambda*I) = det((A - lambda*I)^T) = det(A^T - lambda*I), so A and A^T have identical characteristic polynomials and hence same eigenvalues.