Cramer's Rule

x_i = (A_i)/ (A) — solve Ax = b with three determinants.

The idea

Theorem (Cramer's rule).

Let $A\mathbf{x} = \mathbf{b}$ be a system of $n$ linear equations in $n$ unknowns, and suppose $\det A \neq 0$. Then the system has exactly one solution, and each unknown is a ratio of two determinants: $x_i = \frac{\det(A_i)}{\det(A)},$ where $A_i$ is the matrix $A$ with its $i$-th column replaced by the right-hand side $\mathbf{b}$.

In two variables each determinant measures an area: $\det A$ is the area of the parallelogram spanned by the columns $a_{1}, a_{2}$ of $A$, and $\det(A_x)$ that of the parallelogram with $\mathbf{b}$ in place of $a_{1}$ — so $x$ is a ratio of two areas.

The rule produces one unknown at a time. Elimination solves for all the unknowns together; Cramer's rule writes $x_2$ as an explicit formula in the coefficients without mentioning $x_1$. It is therefore the right tool when only one component of the solution is needed, or when a formula for the solution is more useful than its numerical value.

The hypothesis $\det A \neq 0$ is exactly the condition for $A$ to be invertible, and so exactly the condition for the system to have a single solution. When $\det A = 0$ the formula would divide by zero, and the system has either no solution or infinitely many.

The rule computes a separate determinant for every unknown, so on large systems elimination is much faster. On a $2 \times 2$ system, Cramer's rule is the quicker method.

Ways to work on it

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