Cofactor Expansion: For an $n \times n$ matrix, $\det(A) = \sum_{j=1}^n a_{ij}C_{ij}$ (along row $i$) or $\sum_{i=1}^n a_{ij}C_{ij}$ (along column $j$), where $C_{ij} = (-1)^{i+j}M_{ij}$ is the cofactor, and $M_{ij}$ is the minor (determinant of submatrix after removing row $i$ and column $j$).

Properties:

$\det(A^T) = \det(A)$
$\det(AB) = \det(A)\det(B)$
$\det(kA) = k^n \det(A)$ for an $n \times n$ matrix $A$.
If a matrix has a row/column of zeros, $\det(A)=0$.
If a matrix has two identical rows/columns, $\det(A)=0$.
If one row/column is a scalar multiple of another, $\det(A)=0$.
Swapping two rows/columns changes the sign of the determinant.
Adding a multiple of one row/column to another does not change the determinant.

Inverse of a Matrix

Definition: For a square matrix $A$, its inverse $A^{-1}$ is a matrix such that $AA^{-1} = A^{-1}A = I$.
Existence: $A^{-1}$ exists if and only if $\det(A) \ne 0$. Such a matrix is called non-singular or invertible. If $\det(A)=0$, $A$ is singular.
For $2 \times 2$ Matrix: If $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, then $A^{-1} = \frac{1}{\det(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$.
General Formula: $A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$, where $\text{adj}(A)$ is the adjugate (or classical adjoint) matrix, which is the transpose of the cofactor matrix. $\text{adj}(A) = (C_{ij})^T$.
Properties:
- $(A^{-1})^{-1} = A$
- $(AB)^{-1} = B^{-1}A^{-1}$ (important!)
- $(A^T)^{-1} = (A^{-1})^T$
- $\det(A^{-1}) = \frac{1}{\det(A)}$

Systems of Linear Equations

Matrix Form: A system of $m$ linear equations in $n$ variables can be written as $AX=B$. $$A = \begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn} \end{pmatrix}, \quad X = \begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix}, \quad B = \begin{pmatrix} b_1 \\ \vdots \\ b_m \end{pmatrix}$$
Solution using Inverse: If $A$ is square and invertible ($\det(A) \ne 0$), then $X = A^{-1}B$.
Cramer's Rule: For a system $AX=B$ where $A$ is an $n \times n$ invertible matrix, the solution is given by: $x_j = \frac{\det(A_j)}{\det(A)}$, where $A_j$ is the matrix formed by replacing the $j$-th column of $A$ with the column vector $B$.
Gaussian Elimination / Gauss-Jordan Elimination: Using elementary row operations to transform the augmented matrix $[A|B]$ into row echelon form or reduced row echelon form to solve the system.

Eigenvalues and Eigenvectors

Definition: For a square matrix $A$, an eigenvector $v$ is a non-zero vector that changes at most by a scalar factor when $A$ is applied to it. The scalar factor is called the eigenvalue $\lambda$. $Av = \lambda v$.
Characteristic Equation: To find eigenvalues $\lambda$, solve $\det(A - \lambda I) = 0$. This is a polynomial equation.
Finding Eigenvectors: For each eigenvalue $\lambda_i$, solve the homogeneous system $(A - \lambda_i I)v = 0$ to find the corresponding eigenvectors $v$.
Properties:
- The sum of eigenvalues equals the trace of $A$ (sum of diagonal elements). $\sum \lambda_i = \text{tr}(A)$.
- The product of eigenvalues equals the determinant of $A$. $\prod \lambda_i = \det(A)$.
- Eigenvectors corresponding to distinct eigenvalues are lin