17.4 Evaluating Determinants

Well, from the first two facts alone, we can compute the value of any determinant, and hence the area or volume or whatever of any parallel sided figure.

How?

Well, we can add multiples of rows to one another to get rid of off diagonal elements. When we are done, we can deduce the value of the determinant as the product of the diagonal elements.

Actually, we need only make the elements on one side of the diagonal into 0, and take the product of the diagonal elements. Getting rid of the others is sometimes a nice thing to do, but will not affect the diagonal elements at all.

Let's evaluate the determinants of the following arrays:

If we subtract 5 times the first row from the second in the first matrix we get (0, -3) for the second row, so the determinant is -3. In the second array we subtract times the first row from the second and get (0, 2) as new second row. The determinant of the second matrix is therefore 2 * 2 or 4.

This tells us, by linearity that the determinant of

is -3 + 4 or 1. We can verify this by subtracting of the first row from the second.row, turning that second row into , and the product of the diagonal elements is 1.

This same procedure for evaluating determinants (which is sometimes called "row reduction" and sometimes called "Gaussian elimination") can be applied to square arrays of any size. It is easy to do for 2 by 2 arrays, but it is quite easy to make a mistake even for such. It is still reasonably easy for 3 by 3's but most people will make some silly mistake along the way and get it wrong most of the time. Even you and I can expect to get 4 by 4 determinants wrong most of the time when doing it by hand by this approach, because the steps are so straightforward and so boring. Your mind will stray along the way and you stand an excellent chance of screwing up.

Is this the only way to evaluate a determinant?

No, there are at least two other ways, one of which is equally boring and prone to your making errors. The other is magical and great fun, but surprisingly it is never taught, and practically nobody not reading this has ever heard of it.

The standard approach is to write a formula for the results of the method just described. If you start with rows (a, b) and (c, d). To turn the c into 0 you subtract times the first row from the second. The resulting diagonal elements are then a and and their product is ad - bc. This is the formula for the determinant of a general two by two array.

For three by threes, you will get products of elements one from each row and one from each column (as you must since each such product must be linear in each row and column). There are six such products, and they all occur with appropriate signs.
Applying this formula is also an error prone operation, if done by hand.

Exercise 17.6 Evaluate the following determinants by any method above.

So what is the magical approach?