what is cramers rule?

Question

how do i use it in matrices

Answer 1

Cramer's rule is a Theorem that gives the solution to any system of n linear equations with n unknowns.

 

Let's say you have a system of equations:

 

a_k1x₁ + a_k2x₂ + … + a_knx_n = b_k

 

for k = 1,…,n.

 

We can write this in matrix form as Ax=b where

 

A = [a_ij] is an n×n square matrix.

 

b = [b_i] and x = [x_j] are n-dimensional column vectors.

 

Cramer's rule says that if you let A_i be the matrix formed by replacing the i^th column of A with the vector b, and if det(A)≠ 0 then the unique solution to the system is:

 

x_i = det(A_i) / det(A).

 

Cramer's rule can also be used to get some info when det(A) = 0.

 

In this special case, if det(A_i) = 0 for all i=1,…,n, then there are infinitely many solutions. Otherwise there are no solutions.