1. **Matrices Basics:** A matrix is a rectangular array of numbers arranged in rows and columns. 2. **Matrix Notation:** A matrix with $m$ rows and $n$ columns is called an $m \times n$ matrix. 3. **Matrix Addition and Subtraction:** Matrices of the same size can be added or subtracted by adding or subtracting corresponding elements. 4. **Matrix Multiplication:** If $A$ is an $m \times n$ matrix and $B$ is an $n \times p$ matrix, their product $AB$ is an $m \times p$ matrix where each element is computed as $$ (AB)_{ij} = \sum_{k=1}^n A_{ik} B_{kj} $$ 5. **Identity Matrix:** An identity matrix $I_n$ is an $n \times n$ square matrix with 1's on the diagonal and 0's elsewhere. 6. **Inverse Matrix:** A square matrix $A$ has an inverse $A^{-1}$ if $AA^{-1} = A^{-1}A = I$. 7. **Solving Systems of Linear Equations:** A system can be written as $AX = B$ where $A$ is the coefficient matrix, $X$ is the vector of variables, and $B$ is the constants vector. 8. **Method 1: Gaussian Elimination** - Use row operations to reduce $A$ to row echelon form. - Back-substitute to find the solution vector $X$. 9. **Method 2: Matrix Inverse** - If $A$ is invertible, solve by $X = A^{-1}B$. 10. **Method 3: Cramer's Rule** (for $n \times n$ systems) - Solution for variable $x_i$ is $$ x_i = \frac{\det(A_i)}{\det(A)} $$ where $A_i$ is $A$ with the $i$-th column replaced by $B$. 11. **Important Notes:** - A system has a unique solution if $\det(A) \neq 0$. - If $\det(A) = 0$, the system may have infinite or no solutions. This cheat sheet summarizes key concepts and methods for matrices and solving linear systems.