Cofactor: Definition & Formula

In this lesson, we’ll use step-by-step instructions to show you how to how to find the cofactor of a matrix. We’ll begin with the definition of a cofactor, after which you’ll learn how to use the formula and perform your own calculations.
What Is a Cofactor?
Have you ever used blinders? If so, then you already know the basics of how to create a cofactor. Blinders prevent you from seeing to the side and force you to focus on what’s in front of you. A cofactor is the number you get when you remove the column and row of a designated element in a matrix, which is just a numerical grid in the form of a rectangle or a square. The cofactor is always preceded by a positive (+) or negative (-) sign, depending whether the element is in a + or – position.

This mathematical concept may sound more complicated than it is, so let’s look at an example.

Finding the Cofactor
Let’s start with the matrix:

3-by-3 matrix

Now, how would you go about finding the cofactor of 2? Well, in order to do that, you’d put blinders around the 2 and eliminate the row and column that contain the 2, as shown here:

the cofactor or 2

Once you have the new matrix that doesn’t include 2, you can calculate the determinant, or number derived from a square matrix. You can find the determinant by multiplying the diagonal numbers on the matrix. For example:

3 x 9 = 27
6 x 4 = 24
Next, subtract the value of the second diagonal from the value of the first diagonal: 27 – 24 = 3. Our determinant equals 3.

Lastly, check the sign assigned to the element. Each 3 x 3 determinant has a cofactor sign determined by the location of the element that was eliminated.

First, let’s look at the signs of a 3 x 3 matrix:

signs of a 3-by-3 matrix

Now, let’s locate the original position of the 2. Notice that the + sign is the original location of the 2. Take that + sign, and place it in front of the determinant. The result is +3, or just 3.

Cofactor of a Matrix
If we calculate the cofactor of each element, we can create the cofactor of the matrix.

Let’s return to our matrix:

3-by-3 matrix

In order to calculate the cofactor of the matrix, we need to calculate the cofactors of each element. First, let’s find the cofactor of 3.

cofactor of 3

Once you’ve arrived at your new matrix, calculate the determinant:

2 x 9 = 18
8 x 1 = 8
Subtract the value of the second pair from the value of the first pair, or 18 – 8 = 10. Our determinant equals 10.

Once again, determine the sign dictated by the location of the element you eliminated. In this case, the sign is +, so we would use 10.

Next, let’s find the cofactor of 5.

cofactor of 5

Calculate the determinant: 63 – 32 = 31.

Check the sign determined by position: –

The cofactor of 5 is -31.

Next, let’s find the cofactor of 6.

cofactor of 6

Calculate the determinant: 7 – 8 = -1.

Check the sign determined by position: +

The cofactor of 6 is -1.

Then, find the cofactor of 7.

cofactor of 7

Calculate the determinant: 45 – 6 = 39.

Check the sign determined by position: –

The cofactor of 7 is -39.