Matrices

Basic Operations

Matrix Addition and Subtraction

Scalar Multiplication

Matrix Multiplication (Non-commutativity)

Transpose properties

Trace of a Matrix

Systems of Linear Equations

Augmented Matrix representation

Gaussian Elimination (Row Operations)

Row Echelon Form (REF)

Reduced Row Echelon Form (RREF)

Rank of a Matrix

Free Variables vs Pivot Variables

Determinants

Calculation for $2\times 2$

Sarrus Rule for $3\times 3$

Laplace Expansion (Cofactors)

Properties of Determinants

Relation to Invertibility

Inverse Matrices

Condition for Invertibility ($\det (A)\ne 0$)

Calculation via Adjugate

Calculation via Gauss-Jordan elimination $[A|I]\to [I|A^{-1}]$