2. Consider a 4 x 4 matrix A = [ ], ij $ 4. (a) Write the determinant explicitly. (b) Suppose A = [aj] =4 -|i- j). Find the Cholesky decomposition. Use that to find the determinant. Compare that to the determinant in part (a). 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 Cj = 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 Cj > 0. j = 1. ....n. Hint: use the LU decomposition without pivoting.]