For the first question, write in function of and . For the second question, take an eigenvector for the eigenvalue .
We say a symmetric matrix A is positive definite if Ax dot x>0 for all x not zero, negative definite if Ax dot x<0 for all x not zero and positive semi-definite if Ax dot x >/= 0 for all x.
Show that if A and B are positive definite, then so is A+B
Show that if A is positive definite if and only if all its eigenvalues are positive.