TF. In parts (a)(l) determine whether the statement is true or false, and justify your

Chapter 7, Problem T1

(choose chapter or problem)

TF. In parts (a)(l) determine whether the statement is true or false, and justify your answer. (a) If all eigenvalues of a symmetric matrix A are positive, then A is positive definite. (b) x2 1 x2 2 + x2 3 + 4x1x2x3 is a quadratic form. (c) (x1 3x2)2 is a quadratic form. (d) A positive definite matrix is invertible. (e) A symmetric matrix is either positive definite, negative definite, or indefinite. (f ) If A is positive definite, then A is negative definite. (g) x x is a quadratic form for all x in Rn. (h) If A is symmetric and invertible, and if xT Ax is a positive definite quadratic form, then xT A1x is also a positive definite quadratic form. (i) If A is symmetric and has only positive eigenvalues, then xT Ax is a positive definite quadratic form. ( j) If A is a 2 2 symmetric matrix with positive entries and det(A) > 0, then A is positive definite. (k) If A is symmetric, and if the quadratic form xT Ax has no cross product terms, then A must be a diagonal matrix. (l) If xT Ax is a positive definite quadratic form in two variables and c = 0, then the graph of the equation xT Ax = c is an ellipse. Find an orthogonal matrix P such that PT AP is diagonal. A = 2111 1 211 1 1 2 1 111 2