Cramer’s Rule

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Section 3.2 Cramer’s Rule

Given a matrix equation $A\mathbf{x}=\mathbf{b}\text{,}$ where $A$ is an $n \times n$ matrix. Let $A_i(\mathbf{b})$ be the matrix obtained from $A$ by replacing column $i$ by the vector $\mathbf{b}\text{.}$

\begin{equation*} \begin{array}{llcll} A_i(\mathbf{b})=[\mathbf{a}_1 \amp \cdots \amp \mathbf{b} \amp \cdots \amp \mathbf{a}_n]\\ \amp\amp\uparrow\amp\amp\\ \amp\amp\text{col } i\amp\amp \end{array} \end{equation*}

Theorem 3.2.1.

Let $A$ be an invertible $n \times n$ matrix. For any $\mathbf{b}$ in $\mathbb{R}^n\text{,}$ the unique solution $\mathbf{x}$ of $A \mathbf{x}=\mathbf{b}$ has entries given by

\begin{equation*} x_i=\frac{\operatorname{det} A_i(\mathbf{b})}{\operatorname{det} A}, \quad i=1,2, \ldots, n \end{equation*}

Example 3.2.2.

Use Cramer’s Rule to solve the system of linear equations.

\begin{align*} 2x_1-x_2\amp=5\\ 3x_1-5x_2\amp=11 \end{align*}

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