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Finding the Determinant of a 3x3 Matrix

Instructor: Sharon Linde
What is a determinant and how do you find it? This lesson explains what a determinant is and shows you a step-by-step process for finding the determinant of a 3 x 3 matrix.

Setting Up the Problem

We are about to go over how to find the determinant for a 3 x 3 matrix, but first we'll need to know what a determinant is. A determinant is a single specific number associated with a specific square matrix. We should note that determinants are only defined for square matrices. Let's take a look at the process used to find the determinant for a specific matrix.

  • Step 1 - Write the matrix

We have to know what we're working on, right? Well, here is the 3 x 3 matrix we are going to be using for this exercise.

3 x 3 Matrix
matrix

The 3 x 3 refers to the number of rows and columns in our matrix. Since it has three rows and three columns, we call it a 3 x 3 matrix. Since the number of columns and rows are equal, this is a square matrix - which means that it will have a determinant.

  • Step 2 - Write the matrix with determinant symbols

Determinant of a 3 x 3 Matrix
matrix determinant

There is only a small difference in this image and the last one: the brackets have turned into straight lines. Mathematically speaking, however, this indicates a very large difference. The matrix represents a whole series of relationships between numbers while the determinant is just a single number.

  • Step 3 - Write the matrix without brackets or determinant symbols

Now that we know the matrix we are working on, what a determinant is and how it's written - we can start the process of finding the determinant. This step involves just writing the columns and numbers without any other symbols.

3 Columns of 3 x 3 matrix
matrix determinant

Simple, right? Now let's keep going to the next step.

  • Step 4 - Add the first two columns to the right

Now, to the right of our 3 columns we are going to add two more columns. Not just any columns though - we are simply going to repeat the first two columns from our matrix.

3 Columns of 3 x 3 matrix, plus first two columns repeated
matrix determinant

The dotted line in this picture is just for demonstration purposes - it's not necessary to put this in when you are working on other determinant problems. Although if it helps you keep track of where you are in the process, you can certainly keep it.

  • Step 5 - Add multiplications of first down diagonal

Look at the image below before we talk any further about this step.

First Down Diagonal Multiplication
matrix determinant

Start with the number in the first row and first column and multiply together the three numbers in the diagonal going down and to the right. In the image above these three numbers are circled. We are going to add the numbers from the down diagonals together.

  • Step 6 - Add multiplications of second and third down diagonals

Repeat Step 5 for the second and third down diagonals. Again, we are choosing the three numbers from our extended matrix that are on a diagonal that goes down and to the right. Once we have these numbers, we are going to multiply them together and add them to our growing expression. Don't multiply the numbers at this point - we'll do that later on.

Second and Third Down Diagonal Multiplication
matrix determinant

  • Step 7 - Subtract multiplications of up diagonals

Now we are going to do a similar process with the up diagonals. For each diagonal, we are still going to be choosing three numbers to multiply, but for the up diagonals, we are going to be subtracting the multiplication terms instead of adding.

Up Diagonal Multiplication
matrix determinant

Do you notice how we have a large subtraction sign before each of the up diagonal terms? It's very easy to get the wrong sign in this step, so make sure you know why it's there.

  • Step 8 - Compute results

Now that we have all of the terms for computing our determinant, we can start doing the operations. Remember, a negative times a negative is a positive, and if any of the multipliers are 0, then that term is going to be equal to 0.

What we have after all of the above steps is:

Determinant of A

= +(1)(3)(2) + (-4)(-1)(2) + (0)(0)(0) - (2)(3)(0) - (0)(-1)(1) - (2)(0)(-4)

When we evaluate each of the multiplication terms we get:

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