$\displaystyle A = \left[ \begin{array}{cccc} a & c \\ c & b \end{array} \right] $

Show that if A is positive definite, then both a & b are positive.

I think I'm on the right track with the following...

$\displaystyle z^TAz = \left[ \begin{array}{cccc} Z_1 & Z_2 \end{array} \right]\left[ \begin{array}{cccc} a & c \\ c & b \end{array} \right]\left[ \begin{array}{cccc} Z_1 \\ Z_2 \end{array} \right]$

$\displaystyle = \left[ \begin{array}{cccc} Z_1 & Z_2 \end{array} \right]\left[ \begin{array}{cccc} aZ_1 + cZ_2 \\ cZ_1 + bZ_2 \end{array} \right]$

$\displaystyle = Z_1(aZ_1 + cZ_2) + Z_2(cZ_1 + bZ_2)$

$\displaystyle = aZ_1^2 + 2CZ_1Z_2 + bZ_2^2$

$\displaystyle aZ_1^2 + 2CZ_1Z_2 + bZ_2^2 > 0$

Can anyone point me in the right direction? Thanks.