Prove that determinant of a 2x2 symmetric positive definite matrix is positive by "completing the square" method.

Question

From my understanding, determinant = product of Eigen values. Since it is a positive definite matrix, the eigen values are positive and hence, the determinant is positive.

But the text book is asking about how to prove this using "completing the square" method.

Any help is appreciated. As reference, the book points to this text (for the completing the square method):

Answer 1

If $A=\begin{bmatrix}a & b\\ b & c\end{bmatrix}$ then $\det(A)=ac-b^2$.

1. Take $x=\begin{bmatrix}1 \\ 0\end{bmatrix}$ to prove that $a=x^TAx>0$ (since $A$ pos. def.).
2. Take $A_k=a$ and apply the lemma to get $C-BA_k^{-1}B^T=c-\frac{b^2}{a}>0$.
3. Multiply $a(c-\frac{b^2}{a})=ac-b^2=\det(A)>0$.