rag/knowledge_base/matrices_determinants.md

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
• $\det(A^{-1}) = \frac{1}{\det(A)}$