**Operations on matrices** are the fundamental ways of combining and transforming matrices. The main operations include:
- **Addition**: $A + B$ — entrywise sum (requires same size)
- **Scalar multiplication**: $cA$ — multiply each entry by scalar $c$
- **Matrix multiplication**: $AB$ — row-by-column dot products (requires compatible dimensions)
- **Transpose**: $A^T$ — flip rows and columns
- **Conjugate transpose (adjoint)**: $A^*$ or $A^H$ — transpose + complex conjugate
- **Inverse**: $A^{-1}$ — matrix that satisfies $AA^{-1} = I$ (for nonsingular matrices)
- **Row operations**: elementary operations used in Gaussian elimination
These operations make the set of $m \times n$ matrices into a vector space, and the set of $n \times n$ matrices into a ring.