Cofactor: Definition & Formula

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  • 0:03 What Is a Cofactor?
  • 0:47 Finding the Cofactor
  • 2:02 Cofactor of a Matrix
  • 4:49 Using the Cofactor of…
  • 5:15 Lesson Summary
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Lesson Transcript
Instructor
Kimberly Hopkins
Expert Contributor
Alfred Mulzet

Dr. Alfred Kenric Mulzet received his Ph.D. in Applied Mathematics from Virginia Tech. He currently teaches at Florida State College in Jacksonville.

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.

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Additional Activities

Cofactors and Determinants:

In this exercise, we will see how a judicious choice of row expansion can make calculating the determinant of a matrix much easier. Consider the matrix

{eq}A = \begin{bmatrix} 3 & 2 & 5 \\ 4 & 0 & 1 \\ -3 & 0 & -6 \end{bmatrix} {/eq}

1. Find the cofactor for each entry in the first row.

Answer:

{eq}A_{1, 1} = \begin{bmatrix} 0 & 1 \\ 0 & -6 \end{bmatrix} \\ \\ A_{1, 2} = \begin{bmatrix} 4 & 1 \\ -3 & -6 \end{bmatrix} \\ \\ A_{1, 3} = \begin{bmatrix} 4 & 0 \\ -3 & 0 \end{bmatrix} {/eq}

2. Calculate the determinant of each cofactor:

Answer:

{eq}\mathrm{det} \: A_{1, 1} = (0)(-6) - (1)(0) = 0 \\ \mathrm{det} \: A_{1, 2} = (4)(-6) - (1)(-3) = -21 \\ \mathrm{det} \: A_{1, 3} = (4)(0) - (0)(-3) = 0 {/eq}

3. Now calculate the determinante of {eq}A {/eq} using the cofactors.

Answer:

{eq}\mathrm{det} \: A = 3(0) - 2(-21) + 5(0) = 42. {/eq}

4. We will now calculate the determinant by using the second column. Before we get started, remember that if we expand along the second column, then the determinant is

{eq}\mathrm{det} \: A = a_{1, 2} \: \mathrm{det} A_{1, 2} - a_{2, 2} \: \mathrm{det} A_{2, 2} + a_{3, 2} \: \mathrm{det} A_{3, 2}. {/eq}

Now notice that {eq}a_{2, 2} = a_{3, 2} = 0. {/eq} Therefore we do not need to calculate the determinant of their corresponding cofactors. We already know that

{eq}A_{1, 2} = \begin{bmatrix} 4 & 1 \\ -3 & -6 \end{bmatrix} {/eq}

and

{eq}\mathrm{det} \: A_{1, 2} = (4)(-6) - (1)(-3) = -21. {/eq}

Since {eq}a_{1, 2} = 2, {/eq} this gives us

{eq}\mathrm{det} \: A = -a_{1, 2} \mathrm{det} A_{1, 2} = -2(-21) = 42. {/eq}

Much easier!

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