Cramer's Rule: A Simple Way to Solve System of Linear Equations

Linear algebra is an essential branch of mathematics that deals with the study of linear equations. These equations have numerous applications in various fields such as physics, engineering, economics, and computer science. One of the most significant challenges in solving systems of linear equations is the number of variables involved. However, with cramer's rule , you can simplify the process and find the unknown values with ease.

What is Cramer's Rule?
Cramer's rule is a method used to solve systems of linear equations with multiple unknowns. It was first introduced by Gabriel Cramer, a Swiss mathematician, in the 18th century. The rule states that if we have a system of n linear equations with n unknowns, we can find the values of the unknowns using the determinants of the coefficient matrix and the matrices formed by replacing each column with the constant terms.

How to Use Cramer's Rule
To apply Cramer's rule, you need to follow these steps:

1. Write the system of linear equations in the standard form: Ax = b 2. Find the determinant of the coefficient matrix A. This is denoted as |A|. 3. Create a matrix by replacing the first column of A with the constant terms. This is denoted as A1. 4. Find the determinant of A1. This is denoted as |A1|. 5. Repeat step 4 by replacing the second column with the constant terms and find its determinant. This is denoted as |A2|. 6. Continue this process until you have replaced all the columns with the constant terms. 7. The value of the first unknown (x1) can be found by dividing |A1| by |A|. 8. The value of the second unknown (x2) can be found by dividing |A2| by |A|. 9. The value of the third unknown (xn) can be found by dividing the determinant of the matrix formed by replacing the nth column with the constant terms by |A|. 10. Repeat this process for all the unknowns.

Example
Suppose we have the following system of linear equations:

2x + 3y - z = 5
x - 2y + 4z = -3
x + y + 2z = 4

To solve this system using Cramer's rule, we need to find the determinants of the coefficient matrix and the matrices formed by replacing each column with the constant terms.

|A| = 2 × 4 - 3 × 1 = 8 - 3 = 5
|A1| = 3 × 4 - 2 × 1 = 12 - 2 = 10
|A2| = 2 × 1 - 4 × 1 = 2 - 4 = -2
|A3| = 2 × 1 + 1 × 1 = 2 + 1 = 3

Now, we can find the values of the unknowns:

x1 = |A1| / |A| = 10 / 5 = 2
x2 = |A2| / |A| = -2 / 5
x3 = |A3| / |A| = 3 / 5

Conclusion
Cramer's rule is a powerful tool for solving systems of linear equations. It is a simple and efficient method that can be used to find the values of the unknowns. By following the steps outlined above, you can apply Cramer's rule to solve systems of linear equations with ease. Whether you are a student or a professional, Cramer's rule is an essential tool to have in your mathematical toolbox.

To find more resources on Cramer's rule and linear algebra, you can visit online educational platforms such as Khan Academy, Coursera, and edX. These platforms offer interactive lessons, video lectures, and practice problems to help you master the concepts of linear algebra.
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