Operations on matrices

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.