The Wolfram Language's matrix operations handle both numeric and symbolic matrices, automatically accessing large numbers of highly efficient algorithms. The Wolfram Language uses state-of-the-art algorithms to work with both dense and sparse matrices, and incorporates a number of powerful original algorithms, especially for high-precision and symbolic matrices.

+, *, ^, ... — all automatically work element-wise

Dot (.) — products of matrices, automatically handling row and column vectors

Inverse — matrix inverse (use LinearSolve for linear systems)

MatrixRank — rank of a matrix

NullSpace — vectors spanning the null space of a matrix

RowReduce — reduced row echelon form

PseudoInverse — pseudoinverse of a square or rectangular matrix

Transpose — transpose (, entered with tr)

ConjugateTranspose — conjugate transpose (, entered with ct)

LowerTriangularize, UpperTriangularize — extract the lower- or upper-triangular part of a matrix

Symmetrize — find the symmetric, antisymmetric, etc. part of a matrix

Diagonal — get the list of elements on the diagonal

Tr — trace

Det — determinant

Norm — operator norm, p-norms and Frobenius norm

Adjugate — adjugate

Minors — matrices of minors

Permanent — permanent

KroneckerProduct — matrix direct product (outer product)

MatrixPower — powers of numeric or symbolic matrices

MatrixExp— matrix exponential

MatrixLog — matrix logarithm

MatrixFunction — general matrix function

Eigenvalues, Eigenvectors — exact or approximate eigenvalues and eigenvectors

Eigensystem — eigenvalues and eigenvectors together

CharacteristicPolynomial — symbolic characteristic polynomial