Amy has a master's degree in secondary education and has taught math at a public charter high school.
Watch this video lesson to learn the rules for taking the determinant of any size matrix. Learn the kind of matrix that you need before you can take the determinant. Watch as the pattern of the determinant takes shape.
We define the determinant as a special number calculated from a square matrix. Determinants are important to learn about and be able to calculate because they provide useful information about the matrix, and they help with finding the inverse of the matrix. They are also useful in higher math such as calculus.
When working with determinants, we have a very special symbol for them. Matrices are written with square brackets, while the determinant of a matrix is written with vertical bars, just like the ones you use for absolute value. So the determinant of a matrix M, [M], looks like this: D = |M| where M stands for the matrix M.
The only kinds of matrices for which we can calculate the determinant are square matrices. These are the matrices that have the same number of rows to columns. Yes, the number of rows and columns are the same for a square matrix. They are called square because the shape of the matrix looks like a square. 2 x 2, 3 x 3 and 4 x 4 matrices are all square matrices. We can find determinants for all of these matrices.
The basic determinant is that for the 2 x 2 matrix. The formula for finding the determinant of such a matrix is this:
What is going on here is that we take the numbers of our top row one by one. Looking at the first number on the top row, we visually cancel all the numbers in that same row and column. We are left with the number d. We multiply our a with the d. We do the same for the second number in the top row, b. We visually cancel all the numbers in the same row and column as b. We are left with c. We then multiply these together. Now we subtract this product from the first.
Let's take a look at how this is done with an example. We begin with our 2 x 2 matrix.
The determinant is then written with absolute value bars.
We wrote a capital D to stand for the determinant, and then we have the absolute value bars around our matrix to show that we will be calculating the determinant. The determinant of this matrix is 1 * 4 - 2 * 3. We have taken each number from the top row and multiplied it by its opposite number.
What is actually going on is when we are looking at the first number in the top row, 1, we mentally cancel all the numbers in the same row and column. We are left with the 4. That is the number we multiply our 1 with. Looking at the second number in the top row, the 2, we again mentally cancel all the numbers that are in the same row and column. We mentally cancel the numbers in the top row and in the second column. We are left with the number 3, so we multiply our 2 with 3.
We also have a pattern of plus and minus along the top row. Our first number is always plus, then we have a negative. If our matrix is bigger, then our next number will be a plus, followed by a negative again. That is why we have 1 * 4 - 2 * 3. Making this calculation, we have 4 - 6 for a determinant of -2.
A Higher Order Matrix
If our matrix is larger than 2 x 2, then we call it a higher order matrix. These are the matrices that are 3 x 3, 4 x 4 and so on. We follow the same pattern as we did for finding the determinant of a 2 x 2 matrix to find the determinants of these larger matrices. The formula for the determinant of a 3 x 3 matrix is this:
We are doing the same thing as we did for the 2 x 2 matrix. We take each term of our top row. We visually cancel all the numbers in the same row and column. We multiply the number in the top row with what is left. Since this is a higher order matrix, we are left with another matrix. So we need to take it one step further - we need to take the determinants of the mini matrices that we get. We follow the pattern of plus, minus, plus, minus for putting all the products together.
Let's take a look at how this works with an example. We will find the determinant of this 3 x 3 matrix.
We will work off the top row just like we did for the 2 x 2 matrix. We begin with the first number in the top row: the 1. We mentally cancel all the numbers in the same row and column. We are left with a 2 x 2 matrix where the top row is 5 and 6 and the bottom row is 8 and 9. Hmm. So we are multiplying the 1 with the 2 x 2 matrix. Okay. We will keep it that way for now.
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The next number is the 2. We again mentally cancel all the numbers that are in the same row and column. Again, we are left with a 2 x 2 matrix. This time, the numbers are 4 and 6 in the top row and 7 and 9 in the bottom row. Our third number is 3. We mentally cancel all the numbers that are in the same rows and columns. We are left with a 2 x 2 matrix where the top row is 4 and 5 and the bottom row is 7 and 8. We apply the positive, negative pattern to our top row. Our 1 is positive, our 2 is negative and our 3 is positive.
The little matrices that we've come up with are called our minor matrices. Because they are also square matrices, we can actually go in and repeat the process of finding the determinant with these matrices until we are left with all numbers and no matrices.
Our first minor matrix is the one with 5 and 6 in the top row and 8 and 9 in the bottom row. The determinant of this minor matrix is 5 * 9 - 6 * 8 = 45 - 48 = -3. Good. The next minor matrix is the one with the 4 and 6 on the top and 7 and 9 on the bottom. The determinant of this matrix is 4 * 9 - 6 * 7 = 36 - 42 = -6. The last minor matrix is the one with the 4 and 5 on top and the 7 and 8 on the bottom. The determinant of this matrix is 4 * 8 - 5 * 7 = 32 - 35 = -3.
Evaluating the Determinant
Now that we have the determinants of our minor matrices, we can now plug these in to our original determinant equation to find our answer.
The determinant of our 3 x 3 matrix is 0. We are done!
Do you see that we are simply repeating the same pattern over and over again? We can apply this pattern to any size square matrix.
Now, let's review what we've learned. We've learned that the determinant is a special number calculated from a square matrix. Square matrices look square in shape and have the same number of rows to columns. Higher order matrices are matrices that are larger than 2 x 2. So, 3 x 3, 4 x 4 and so on are all higher order square matrices.
The symbol for determinant is the absolute value bars. The process of finding the determinant is to work off the top row. We first take the first number in the top row, mentally cancel all the numbers in the same row and column as that number, and then we multiply what is left with that number.
We repeat this process with the rest of the numbers in the top row. The top row also follows a positive, negative pattern. The first number is always positive, the next one is negative and then it repeats. So, the determinant of a 2 x 2 matrix with 1 and 2 in the top row and 3 and 4 in the bottom row is 1 * 4 - 2 * 3 = 4 - 6 = -2.
For larger matrices, you will end up with minor matrices, smaller matrices after you cancel your row and column. After making the first round of determinant calculations, you go back in and find the determinant of these minor matrices. You keep repeating this process over and over until you reach the point where you are left with just numbers. Then you finish evaluating, and you are done!
Upon completing this lesson, you will be able to:
Discuss determinants, higher order matrices and minor matrices
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