Determinants and Cramer's Rule

In summary, determinants are mathematical values used to solve systems of equations and determine the number of solutions. They can be calculated using a specific formula based on the size of the matrix or by using Cramer's rule, which involves creating a matrix of coefficients and constants. Cramer's rule is useful for small systems and avoids complex calculations, but it has limitations and becomes inefficient for larger systems.
  • #1
rocomath
1,755
1
I'm trying to learn about Determinants and Cramer's Rule.

If a multiple of one row is added to another row, the value of the determinant is not changed. This applies to columns, also.

15 14 16
18 17 32
21 20 42

Factoring a 3 from C1 and a 2 from C3 =

6 times

5 14 13
6 17 13
7 20 21

Now subtracting C3 from C2 =

6 times

5 1 13
6 1 16
7 -1 21

So, C3-C2. Why isn't C2 ...

13-14 = -1
16-17 = -1
21-20 = 1

I notice it's just opposite signs, but where does the -1 multiple come from.
 
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  • #2
Uh! Ok it says "subtract the third column from the second column" meaning C2-C3.
 
  • #3


Cramer's Rule is a method for solving systems of linear equations using determinants. It is based on the fact that the determinant of a matrix represents the volume of a parallelepiped formed by the column vectors of the matrix. When a multiple of one row (or column) is added to another row (or column), the volume of the parallelepiped remains the same, but the shape changes. This is why the value of the determinant is not changed.

In the given example, when we factor a 3 from C1 and a 2 from C3, we are essentially changing the shape of the parallelepiped, but its volume remains the same. This is why the determinant remains unchanged.

When we subtract C3 from C2, we are essentially removing the third column from the parallelepiped. This results in a new shape with a different volume, but the determinant remains the same.

The -1 multiple in C3-C2 comes from the fact that when we subtract C3 from C2, we are essentially flipping the third column (since we are subtracting it instead of adding it). This results in the opposite signs in the values of the third column. This is why the -1 multiple appears in the resulting matrix.

In summary, determinants and Cramer's Rule are important concepts in linear algebra that help us solve systems of linear equations. Understanding the relationship between determinants and the volume of parallelepipeds is key to understanding Cramer's Rule and its applications.
 

1. What are determinants?

Determinants are mathematical values that are used to solve systems of equations and determine whether a system has a unique solution, no solution, or infinite solutions.

2. How are determinants calculated?

Determinants are calculated by using a specific formula based on the size of the matrix. For a 2x2 matrix, the determinant is calculated by multiplying the values in the main diagonal and subtracting the product of the values in the other diagonal. For larger matrices, the determinant can be found by expanding along a row or column and using the cofactor method.

3. What is Cramer's rule?

Cramer's rule is a method for solving a system of linear equations using determinants. It involves creating a matrix of the coefficients of the variables and a column matrix of the constants. The solution for each variable is then found by dividing the determinant of the variable's column matrix by the determinant of the coefficients matrix.

4. When is Cramer's rule useful?

Cramer's rule is useful when solving systems of equations with a small number of variables (typically 2 or 3). It can also be helpful when solving systems with fractions or irrational numbers, as it avoids the need for complex calculations.

5. Are there any limitations to using Cramer's rule?

Yes, there are limitations to using Cramer's rule. It can only be used to solve systems of equations with the same number of equations as variables. It also becomes computationally inefficient for larger systems, as the number of determinants that need to be calculated increases with the size of the system.

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