Matrices, Determinants & Cramer's Rule 📊

### Matrices: The Mathematical Backbone of Computer Graphics and AI! 💻 A **Matrix** is a rectangular array of numbers arranged in rows and columns: $$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ **1. Determinant of a $2 \times 2$ Matrix**: $$\det(A) = |A| = ad - bc$$ * A matrix is **singular** if $\det(A) = 0$ (it has no inverse!). **2. Inverse of a $2 \times 2$ Matrix**: $$A^{-1} = \frac{1}{|A|} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$ **3. Cramer's Rule for Linear Systems**: For the system: $$a_1 x + b_1 y = c_1$$ $$a_2 x + b_2 y = c_2$$ $$x = \frac{\Delta_x}{\Delta}, \quad y = \frac{\Delta_y}{\Delta}$$ Where $\Delta = |A|$, $\Delta_x$ is formed by replacing the $x$-column with constants, and $\Delta_y$ by replacing the $y$-column with constants!