3. Consider a real symmetric n x n matrix, A = AT. (a) Show that a necessary condition to be positive definite is that all diagonal ele- ments are positive. Show that this condition is not sufficient. (b) Consider the sequence of determinants Cf = det Aj, where Aj is obtained by removing all rows and columns from A where the column and row indices are greater than j. Show that a necessary and sufficient condition for A to be positive definite is that all C, > 0, j = 1,...,n. Hint: use the LU decomposition without pivoting.]