After reading the previous section, you might have wondered if you can evaluate 4 x 4 determinants, 5 x 5 determinants, and larger determinants using the method for evaluating 3 x 3 determinants. And the answer is, no, we need a more complex tool in our toolbelt to evaluate such determinants. And that tool is called "evaluating by minors."

We'll begin by showing you how to evaluate a 3 x 3 determinant by minors, and then on the next page we'll extend it to 4 x 4 determinants.

Before we even get started, I want to show you a 3 x 3 matrix, which we will call our "sign" matrix:

S =

This is an easy matrix to make; just start with "+1" in the upper left corner, and then make sure that wherever you have "plusses" you have "minusses" next to them (both horizontally and vertically). We'll make use of this in a minute. (Once you get used to evaluating by minors, you won't even write out this matrix; you'll automatically know which sign goes with which entry.)

Let's take the following determinant, and use it as an example of how to evaluate by minors:

When evaluating by minors, we pick a row or a column to use in the evaluation process. In our case, let's pick the first column. We could just have easily picked the second row, for example, or the third column.

The process we do next, we will do for each of the three numbers in the first column.

First Row 
First, we will cross out everthing in the first row and the first column:

What remains is a 2 x 2 determinant: .

We will multiply this by the number in the first row, first column of our determinant, and also by the number in the first row, first column of our sign matrix: (+1)(1) = -6

Second Row 
Now we cross out everything in the second row and first column:

What remains is a 2 x 2 determinant: .

We will multiply this by the number in the second row, first column of our determinant, and also by the number in the second row, first column of our sign matrix: (-1)(4) = 28

Third Row 
Now we cross out everything in the third row and first column:

What remains is a 2 x 2 determinant: .

We will multiply this by the number in the third row, first column of our determinant, and also by the number in the third row, first column of our sign matrix: (+1)(2) = 8

Adding it Up
Now we just combine these three numbers: -6 + 28 + 8 = 30

To put the entire process in one single calculation, it would look like this:

1 - 4 + 2

Benefit to Using Minors
You might think "This is much more complicated that evaluating by diagonals...why would we even do this?" Trust me when I say there are reasons why this method is useful. One reason is that 4 x 4 determinants can't be evaluated without this method (we'll look at this on the next page). The other reason is that evaluating by minors can make the process much simpler. Here's an example:

Evaluate

When I look at this determinant, I notice right away that the second row is mostly zeroes. Do you realize what this means? It means that if I expand the determinant around the second row, two of the three minors are going to get multiplied by zero...which means I don't even need to evaluate them!

= (-1)(0) + (1)(0) + (-1)(1) = (-1)(1)(26) = -26.

As a matter of fact, even though I wrote out the whole process, I really did the problem in my head, because I just had to focus on that last two by two determinant, which can be easily evaluated mentally.

1.

If you were evaluating by minors, which row or column would be the most sensible one to evaluate around, and why?

2.

Evaluate by minors.

3.

Evaluate by minors.

4.

Evaluate by minors.