Chapter 3.1, Problem 18E

The expansion of a 3 × 3 determinant can be remembered by the following device. Write a second copy of the first two columns to the tight of the matrix, and compute the determinant by multiplying entries on six diagonals:

Add the downward diagonal products and subtract the up-ward products. Use this method to compute the determinants in Exercises 15–18. Warning: This trick does not generalize in any reasonable way to 4 × 4 or larger matrices.

18. $\left|\begin{array}{ccc}1& 3& 4\\ 2& 3& 1\\ 3& 3& 2\end{array}\right|$