Showing matrix $\left[\begin{smallmatrix} 4 & 1 & 1\\1 & 2 & -1 \\ 1 & -1 & 3 \end{smallmatrix}\right]$ is positive definite $$\begin{bmatrix}
    x_1 & x_2 & x_3
\end{bmatrix}\begin{bmatrix}
    4 & 1 & 1\\1 & 2 & -1 \\ 1 & -1 & 3
\end{bmatrix}\begin{bmatrix}
    x_1\\x_2\\x_3
\end{bmatrix} = \\4x_1^2 + x_1x_2 + x_3x_1 + x_2x_1 + 2x_2^2 -x_2x_3 + x_1x_3-x_2x_3+3x_3^2$$
which I cannot put in a form that is $>0$. So how can this matrix be positive definite? 

Note that we can show that a matrix is positive definite by looking at its $n$ upper left determinants. Note that for the matrix 
$$\begin{bmatrix}
    4 & 1 & 1\\1 & 2 & -1 \\ 1 & -1 & 3
\end{bmatrix}$$
We have 
$$\begin{vmatrix}
4
\end{vmatrix} = 4$$
and
$$\begin{vmatrix}
4 & 1\\
1 & 2
\end{vmatrix} = 7$$
and lastly 
$$\begin{vmatrix}
4 & 1 & 1\\
1 & 2 & -1\\
1 & -1 & 3
\end{vmatrix} = 13$$
Since $4,7,13 > 0$ the matrix is positive definite.
