Cramer's Rule

linear-algebra

Definition

Cramer's Rule

Let $A\in F^{n\times n}$ be a regular matrix and $\mathbf{b}\in F^n$. Denote $A_j$ ($1\le j\le n$) the matrix obtained from $A$ by replacing its $j$-th column with $\mathbf{b}$. Then, the unique solution of the linear system $A\mathbf{x}=\mathbf{b}$ is given by:

$$\mathbf{x}=\frac{1}{\det A}\cdot \left[\begin{array}{c}\det A_1\\ ⋮\\ \det A_n\end{array}\right]$$

where $x_j$ is the $j$-th component of the solution vector $\mathbf{x}$.

Thus, every component of the solution vector can be computed by:

$$x_j=\frac{\det A_j}{\det A}$$