Cramer’s rule

Cramer’s rule is used to solve a system of linear equations whose coefficient matrix is non-singular.

Cramer systems

Take a system of linear equations Ax = b. We call this system a Cramer system if A is a square non-singular matrix. This means that the determinant of the matrix is always non-zero, so the matrix has no linearly dependent rows or columns.

By the Rouché–Capelli theorem, every Cramer system has exactly one solution. The Rouché–Capelli theorem says that if rank(A) = rank(A|b) = k, then the system has a solution. The matrix A of a Cramer system is square and has the greatest possible rank. This means that if we attach any further column to this matrix, that column must be a linear combination of the columns of A. So attaching the column b does not increase the rank of A.

The Rouché–Capelli theorem also says that we need n − k parameters to express the solution, where n is the number of variables and k is the rank rank(A|b). For a Cramer system, n = k, so we need zero parameters to express the solution. That is why every Cramer system has exactly one solution.

Next, let A_i be the matrix A modified so that column i is replaced by the column b. Formally, we define the matrix A_i like this:

$$A_i=\left(\begin{array}{ccccccc}{a}_{1,1}& \dots & a_{1,i-1}& b_1& a_{1,i+1}& \dots & a_{1,n}\\ a_{2,1}& \dots & a_{2,i-1}& b_2& a_{2,i+1}& \dots & a_{2,n}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ a_{n,1}& \dots & a_{n,i-1}& b_n& a_{n,i+1}& \dots & a_{n,n}\end{array}\right)$$

where a_ij are the elements of A and b_i are the elements of b.

Cramer’s rule

Let Ax = b be a Cramer system of linear equations. Cramer’s rule says that this system has exactly one solution (x₁, x₂, …, x_n), where

$$x_i=\frac{\text{det }{A}_i}{\text{det }A}=\frac{|A_i|}{|A|}.$$

Let’s see the rule in action on a small example. We solve the following system using Cramer’s rule.

$$\begin{array}{ccccc}{x}_1& +& 2x_2& =& 5\\ 4x_1& +& 3x_2& =& 15& \end{array}$$

The coefficient matrix is:

$$A=\left(\begin{array}{cc}1& 2\\ 4& 3\end{array}\right)$$

The determinant of the matrix is:

$$|A|=3-8=-5.$$

Next we form the matrices A₁ and A₂. They are:

$$A_1=\left(\begin{array}{cc}5& 2\\ 15& 3\end{array}\right),A_2=\left(\begin{array}{cc}1& 5\\ 4& 15\end{array}\right)$$

We work out the determinants of these matrices:

$$\begin{array}{rcl}|A_1|& =& 5\cdot 3-15\cdot 2=-15\\ |A_2|& =& 1\cdot 15-4\cdot 5=-5\end{array}$$

Now we work out x₁ and x₂ in turn. By Cramer’s rule, for x₁ we have

$$x_1=\frac{|A_1|}{|A|}$$

Substituting, we get

$$x_1=\frac{-15}{-5}=3$$

The same for the second element x₂:

$$x_2=\frac{|A_2|}{|A|}=\frac{-5}{-5}=1$$

The solution of the system is therefore the pair (x₁, x₂) = (3, 1).

Cramer’s rule is most useful when you want to turn the method into an algorithm, because it is easy to program. Solving a system by hand with Cramer’s rule is usually more work than solving it by Gaussian elimination.