Matrices

1. Elementary Transformations

Interchange of rows/columns: $R_i \leftrightarrow R_j$ or $C_i \leftrightarrow C_j$
Scalar multiplication: $R_i \rightarrow kR_i$ or $C_i \rightarrow kC_i$ (where $k \neq 0$)
Addition of scalar multiples: $R_i \rightarrow R_i + kR_j$ or $C_i \rightarrow C_i + kC_j$ (where $k \neq 0$)

Definition of Inverse: $AA^{-1} = A^{-1}A = I$
Condition for Inverse: $|A| \neq 0$ (Matrix must be non-singular)
Co-factor of an element: $A_{ij} = (-1)^{i+j}M_{ij}$ (where $M_{ij}$ is the minor)
Adjoint Matrix: $\text{adj } A = [A_{ij}]^T$ (Transpose of the co-factor matrix)
Adjoint Property: $A(\text{adj } A) = (\text{adj } A)A = |A|I$
Inverse by Adjoint Method:
$$ A^{-1} = \frac{1}{|A|}(\text{adj } A) $$

Transpose: If $A = [a_{ij}]_{m \times n}$, then $A^T = [a_{ji}]_{n \times m}$
Symmetric Matrix: $A = A^T$
Skew-symmetric Matrix: $-A = A^T$
Matrix Decomposition (Symmetric & Skew-symmetric parts):
$$ A = \frac{1}{2}[A + A^T] + \frac{1}{2}[A - A^T] $$
Commutative Law of Addition: $A + B = B + A$
Associative Law for Addition: $(A + B) + C = A + (B + C)$
Associative Law of Multiplication: $(AB)C = A(BC)$
Left Distributive Law: $A(B + C) = AB + AC$
Right Distributive Law: $(A + B)C = AC + BC$