Cramer's rule

Cramer's rule is a way of solving a system of linear equations using determinants. Consider the following system of equations:

The above system of equations can be written in matrix form as Ax = b, where A is the coefficient matrix (the matrix made up by the coefficients of the variables on the left-hand side of the equation), x represents the variables in the system of equations, and b represents the values on the right-hand side of the equation:

Given the above, Cramer's rule states that the solution to the system of equations can be found as:

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

Computing the determinant of each of these matrices makes it possible to find the solution for the desired variable. For example, to find x₁, compute the determinant of A₁ then divide it by the determinant of A. There are a number of different ways to compute the determinants of square matrices; refer to the determinant page if necessary. For this example, we will use cofactor expansion to find the determinants:

Thus:

The determinants of A₂ and A₃ can be calculated in the same manner:

The solutions to x₁ and x₂ can then be calculated as:

We can then test the solutions with each of the equations in the system:

The general form of Cramer's Rule, using matrix notation can be written as follows. A system of linear equations,

can be written in the form Ax = b in matrix form as,

,

and Cramer's rule is as stated above,

,

where A_i is the new matrix formed by replacing the ith column of A with the b column vector, and v_i represents the ith column vector in a:

Proof of Cramer's rule

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), where k is a scalar, 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 factor x_i from the ith column which contains x_iv_i. In other words,

=

Since

,

we can combine the above to give 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 doesn't 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.