# Matrices and Determinants

## Matrix Operations
- Addition: $(A+B)_{ij} = a_{ij} + b_{ij}$ (same dimensions required)
- Scalar multiplication: $(kA)_{ij} = k \cdot a_{ij}$
- Matrix multiplication: $(AB)_{ij} = \sum_k a_{ik} b_{kj}$
- $AB \neq BA$ in general (not commutative)

## Special Matrices
- Identity: $AI = IA = A$
- Transpose: $(A^T)_{ij} = a_{ji}$, $(AB)^T = B^T A^T$
- Symmetric: $A^T = A$
- Skew-symmetric: $A^T = -A$
- Orthogonal: $A^T A = I$

## Determinants (2×2 and 3×3)
$\det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$

$\det \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} = a(ei-fh) - b(di-fg) + c(dh-eg)$

## Determinant Properties
- $\det(A^T) = \det(A)$
- $\det(AB) = \det(A) \cdot \det(B)$
- $\det(kA) = k^n \det(A)$ for $n \times n$ matrix
- Swapping two rows/columns: changes sign
- Two identical rows/columns: determinant = 0
- Row/column of zeros: determinant = 0

## Inverse Matrix
$A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$
- Exists only when $\det(A) \neq 0$ (non-singular)
- $(AB)^{-1} = B^{-1}A^{-1}$

## Cramer's Rule
For system $AX = B$ where $\det(A) \neq 0$:
$x_i = \frac{\det(A_i)}{\det(A)}$ where $A_i$ is $A$ with column $i$ replaced by $B$

## JEE Tips
- For $3 \times 3$ determinant: use Sarrus' rule or cofactor expansion
- Singular matrix: $\det(A) = 0$, no unique inverse
- $\det(A^{-1}) = \frac{1}{\det(A)}$