Learn Linear Algebra

Cramers Rule

Let $A$ be an invertible $n \times n$ matrix. For any $\vec{b} \in \mathbb{R}^n$, the unique solution $\vec{x}$ of $A\vec{x} = \vec{b}$ has entries given by $$x_i = \frac{\det A_i(\vec{b})}{\det A}, \quad i = 1, 2, \dots, n$$

An Inverse Formula

Let $A$ be an invertible $n \times n$ matrix. Then

$$A^{-1} = \frac{1}{\det A} \, \text{adj} A$$