How to show this particular $4 \times 4$ matrix is positive definite?

@2rd_7's suggestion to calculate the pivots seems quick and straightforward. You just need to keep subtracting out multiples of the remaining rows (i.e., add the first row to the second, subtract 2 times the first row from the third, etc.) until you get an upper triangluar matrix: $U = \begin{bmatrix} 1 & -1 & 2 & 0\\ 0 & 3 & 1 &1\\ 0 & 0 & 3 & -1\\ 0 & 0 & 0 & 26/9\end{bmatrix}$.

The pivots are now on the diagonal and are all positive, so the matrix is positive definite.