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Linear Systems
Mathematica incorporates the latest algorithms for solving industrial-scale linear systems, automatically switching between optimal dense and sparse algorithms—and handling exact, symbolic, and arbitrary-precision as well as machine-precision computation.
LinearSolve solve a linear system, dense or sparse
LinearSolveFunction a function created to repeatedly solve a linear system
Inverse explicit dense matrix inverse
    
NullSpace vectors spanning the null space of a matrix
MatrixRank rank of a matrix
Det determinant
    
Minors matrices of minors
RowReduce generate a row echelon form
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