Cramer's Rule

Cramer's Rule

Cramer's Rule. Let $Ax = b$ be a system of linear equations for $A \in M_{n \times n}(F)$. If $A$ is invertible, then $Ax = b$ has the unique solution given by $$x_j = \frac{\det C_j}{\det A}, \quad j = 1, ..., n,$$ where $C_j$ is the matrix obtained from $A$ by replacing the $j$-th column with the column vector $b$.

Proof. We have $\textbf{x} = A^{-1} \textbf{b}$. Then $$\textbf{x} = \frac{1}{\det A} \begin{bmatrix} A_{11} & \cdots & A_{n1} \\ \vdots & \ddots & \vdots \\ A_{1n} & \cdots & A_{nn} \end{bmatrix} \begin{bmatrix} b_1 \\ \vdots \\ b_n \end{bmatrix} \\ = \frac{1}{\det A} \begin{bmatrix} A_{11}b_1 + \cdots A_{n1} b_n \\ \vdots \\ A_{1n}b_1 + \cdots + A_{nn}b_n \end{bmatrix}$$ Thus $$x_j = \frac{1}{\det A}(A_{1j}b_1 + \cdots + A_{nj} b_n) = \frac{\det(C_j)}{\det A} \blacksquare$$