Chapter 6.7, Problem 1E

For each of the following matrices, compute the determinants of all the leading principal submatrices and use them to determine whether the matrix is positive definite:

1. $\left(\begin{array}{cc}2& -1\\ -1& 2\end{array}\right)$
2. $\left(\begin{array}{cc}3& 4\\ 4& 2\end{array}\right)$
3. $\left(\begin{array}{ccc}6& 4& -2\\ 4& 5& 3\\ -2& 3& 6\end{array}\right)$
4. $\left(\begin{array}{ccc}4& 2& 1\\ 2& 3& -2\\ 1& -2& 5\end{array}\right)$