Solving a 4x4 Determinant: A Guide for Math Projects

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In summary, The conversation discusses different methods for solving a 4x4 determinant and converting it to its upper triangular form. These methods include applying Gaussian Elimination and the permutation definition, which involves expanding by minors. The link provided explains the process of expanding by minors in more detail. The conversation suggests that it is easier to solve the determinant using the minor expansion method, rather than the permutation definition.
  • #1
jebeagles
I was doing a math project, and I was wondering if anyone knew how to solve a 4x4 determinant.
 
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  • #2
Convert it to it's upper triangular form using Gaussian Elimination and the determinant should be the product of it's diagonal elements. I think.
 
  • #4
Aplying the permutation definition, you will have 24 terms to sum. Its much easier to do by minors.
 
  • #5
The DEFINITION of a genereal n by n determinant is this: form all possible products taking one number from each row and column. There will be n! ways to do this. If you write the terms so that the numbers are in order of the columns, the row numbers will be a permutation of 1,2,3...n. Multiply each product by -1 if this is an odd permutation, 1 if even permutation, and add.

The simplest way to calculate it is row reduce as Jhageb suggested.

Second simplest way is to "expand by minors".
 

1. What is a 4x4 determinant?

A 4x4 determinant is a mathematical calculation used in linear algebra to determine the unique solution of a system of four linear equations. It involves arranging four rows and columns of numbers in a square matrix and performing a specific set of operations to find a single numerical value called the determinant.

2. Why is solving a 4x4 determinant important?

Solving a 4x4 determinant is important because it allows us to find the solution to a system of four linear equations, which is a common problem in many fields such as engineering, physics, and economics. It also helps us understand the properties of a matrix and its relationship to transformations and inverses.

3. What are the steps to solve a 4x4 determinant?

The steps to solve a 4x4 determinant are as follows:

  1. Arrange the numbers in a square matrix with four rows and columns.
  2. Choose a row or column and expand it by factoring out the first term in that row or column.
  3. Multiply each element in the expanded row or column by its corresponding minor, which is the determinant of the matrix formed by removing the current row and column.
  4. Alternate between adding and subtracting these products to find the final determinant.

4. What are some common mistakes when solving a 4x4 determinant?

Some common mistakes when solving a 4x4 determinant include forgetting to alternate between adding and subtracting the products, making errors in the expansion or calculation of minors, and mixing up rows or columns during the process. It is important to carefully follow the steps and double-check calculations to avoid these mistakes.

5. Are there any shortcuts for solving a 4x4 determinant?

Yes, there are some shortcuts for solving a 4x4 determinant, such as using the rule of Sarrus or the Laplace expansion method. However, these methods may not be as efficient or applicable to other matrix sizes, so it is important to have a good understanding of the standard method as well.

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