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How to Evaluate Higher-Order Determinants in Algebra

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  • 0:01 Determinants0
  • 3:05 A Higher Order Matrix
  • 3:58 Minor Matrices
  • 6:02 Evaluating the Determinant
  • 6:24 Lesson Summary
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Instructor: Yuanxin (Amy) Yang Alcocer

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.

Determinants

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:

matrix determinant

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.

matrix determinant

The determinant is then written with absolute value bars.

matrix determinant

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:

matrix determinant

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.

matrix determinant

Minor Matrices

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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