What is Cramer's rule?

Cramer's rule is a method for solving a system of linear equations with the same number of equations as unknowns. It uses determinants to provide a unique solution for each variable.

Applying Cramer's rule

Consider a system of linear equations:

We begin by writing these equations in the form $AX = B$.

Now, calculate the determinant of the coefficient matrix $D$ and the determinants obtained by replacing each column with the constants $D_x, D_y, D_z$:

The solutions for the variables $x$, $y$, and $z$ are then obtained by dividing the determinants $D_x$, $D_y$, and $D_z$ by the main determinant $D$:

Note: Learn how to calculate the determinant of the matrix.

Example

Let's consider a simple example of a $2\times 2$ system of linear equations and apply Cramer's rule to solve for the variables.

Here, we have two equations with two variables, $x$ and $y$. We'll calculate the determinant of the coefficient matrix $D$, and the determinants obtained by replacing each column with the constants $D_x, D_y$:

Now, we use Cramer's rule to find the solutions for $x$ and $y$:

After substituting the determinants:

The solution for the system of equations is approximately:

Conclusion

Cramer's rule finds applications in various fields, such as engineering, physics, and economics. It's commonly used for solving small systems of equations or for educational purposes to illustrate the concepts of determinants and linear equations.