If you thought evaluating 3 x 3 determinants was beastly, wait until you see what it looks like to evaluate 4 x 4 determinants. If you haven't read our page about evaluating 3 x 3 determinants by minors, please skip back a page to read that first.

To begin, we need a "sign matrix" that's a 4 x 4 matrix. You should be able to create that matrix without any help; it's just like the one on the previous page, except it has 4 rows and 4 columns:

Now let's do an example:

Just as we did with the 3 x 3 determinants, we're going to look to see if we have a row or column with one or more zeroes, because that row/column will be the easiest to expand around. It looks like the third column is the one we want; it has 2 zeroes.

And that's it...except, of course, that we now have two 3 x 3 determinants we have to evaluate in order to finish. But you know how to do those, right? You can either evaluate them by minors or diagonals.

= (-1)(1)(3) + (-1)(2)(-1) = -3 + 2 = -1

As determinants increase in size, the evaluation process increases in complexity very quickly. If you wanted to evaluate a 5 x 5 determinant, you would need to pick a row or column to expand around, but the expansion would result in a bunch of 4 x 4 determinants, which you would then need to expand around a row or column, which would result in an even larger bunch of 3 x 3 determinants to evaluate. It's not pretty. And it's why we have computers; they are happy to spend their time grinding out such ugly things for us.