# What is Cramer's Rule?

Nov 12, 2015

It is a rule used to help solve a system of linear equations.

#### Explanation:

Let $A x = b$ be a linear system of n equations in n unknowns, where ${A}_{n \times n}$ is the coefficient matrix of the system.

Let the determinant of the A be $\det \left(A\right) = \Delta$.

Now replace any column ${a}_{j}$ of A with the column $b$, the solution set.

Form the new determinant of the new matrix so formed and call it ${\Delta}_{j}$.

Then by Kramer's Rule, the solution value for the variable ${x}_{j}$ may be given by
${x}_{j} = \frac{{\Delta}_{j}}{\Delta}$.

We may repeat this $\forall j = 1 , 2 , 3 , \ldots . , n$.