Publish Date: June 10, 2024

Cramer's Rules

Cramer’s Rule provides a method for solving systems of linear equations using determinants. It states that given a system of linear equations in variables, if the coefficient matrix is square and has a non-zero determinant, and if is the column vector of constants on the right-hand side of the equations, then the solution vector can be expressed as the ratio of the determinants of matrices obtained by replacing each column of with , divided by the determinant of .

Here’s how we can apply Cramer’s Rule step by step with an example using a 3x3 matrix:

Let’s say we have the system of equations:

We represent this system in matrix form as , where:

First, we find the determinant of matrix :

Now, we find the determinants obtained by replacing each column of with :

Finally, we calculate the solution vector :

So, the solution to the system of equations is .

This is how Cramer’s Rule can be applied step by step to solve a system of linear equations.