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Cramer's rule

Cramer's rule is a way of solving a system of linear equations using determinants.

Suppose we are trying to solve a system of linear equations such that

...

or Ax = b in matrix form, where

Cramer's Rule says that

where A_i is a new matrix formed by replacing the ith column of A with the b vector.

Example

In this case,

By Cramer's rule,

x₁	=	=	=	-5
x₂	=	=	=	3
x₃	=	=	=	-8

We can then check that

Proof

We reinterpret the matrix-vector equation Ax = b as

In other words, b = x₁v₁ + ... + x_nv_n where each v_i is the ith column of matrix A (see Matrix Multiplication). If we plug this expression for b into A_i, the matrix made by replacing the ith column of A with b, we get:

From the, properties of determinants, we can perform column operations of the type (i) + k(j) → (i) without changing the determinant. Therefore we can use the columns containing v₁,..., v_{i - 1}, v_{i + 1},... v_n to subtract out every term in x₁v₁ + ... + x_nv_n except for x_iv_i. In other words,

From another property of determinants, a column of type k(i) → (i) has the same effect of multiplying the determinant by k. Therefore we can pull the scalar fact x_i from the iith column which contains x₁v_i. In other words,

Since

combining (1.) and (2.) gives us det(A_i) = x_idet(A). Dividing by det(A) gives us , which is the original statement of Cramer's rule.

Limitations of Cramer's rule

Because we are dividing by det(A) to get , Cramer's rule only works if det(A) ≠ 0. If det(A) = 0, Cramer's rule cannot be used because a unique solution doesnt exist since there would be infinitely many solutions, or no solution at all.
Cramer's rule is slow because we have to evaluate a determinant for each x_i. When we evaluate each det(A_i) we have to perform Gaussian elimination on each A_i for a total of n times. In comparison, if we were to use the augmented matrix, [A|b], we would only need to perform Gaussian elimination once to solve Ax = b.