> with(LinearAlgebra); [Add, Adjoint, BackwardSubstitute, BandMatrix, Basis, BezoutMatrix, BidiagonalForm, BilinearForm, CharacteristicMatrix, CharacteristicPolynomial, Column, ColumnDimension, ColumnOperation, ColumnSpace, CompanionMatrix, ConditionNumber, ConstantMatrix, ConstantVector, CreatePermutation, CrossProduct, DeleteColumn, DeleteRow, Determinant, DiagonalMatrix, Dimension, Dimensions, DotProduct, Eigenvalues, Eigenvectors, Equal, ForwardSubstitute, FrobeniusForm, GaussianElimination, GenerateEquations, GenerateMatrix, GetResultDataType, GetResultShape, GivensRotationMatrix, GramSchmidt, HankelMatrix, HermiteForm, HermitianTranspose, HessenbergForm, HilbertMatrix, HouseholderMatrix, IdentityMatrix, IntersectionBasis, IsDefinite, IsOrthogonal, IsSimilar, IsUnitary, JordanBlockMatrix, JordanForm, LA_Main, LUDecomposition, LeastSquares, LinearSolve, Map, Map2, MatrixAdd, MatrixInverse, MatrixMatrixMultiply, MatrixNorm, MatrixScalarMultiply, MatrixVectorMultiply, MinimalPolynomial, Minor, Multiply, NoUserValue, Norm, Normalize, NullSpace, OuterProductMatrix, Permanent, Pivot, QRDecomposition, RandomMatrix, RandomVector, Rank, ReducedRowEchelonForm, Row, RowDimension, RowOperation, RowSpace, ScalarMatrix, ScalarMultiply, ScalarVector, SchurForm, SingularValues, SmithForm, SubMatrix, SubVector, SumBasis, SylvesterMatrix, ToeplitzMatrix, Trace, Transpose, TridiagonalForm, UnitVector, VandermondeMatrix, VectorAdd, VectorAngle, VectorMatrixMultiply, VectorNorm, VectorScalarMultiply, ZeroMatrix, ZeroVector, Zip] > A := Matrix(2,2,[[a[1,1], a[1,2]], [a[2,1],a[2,2]]]); [a[1, 1] a[1, 2]] A := [ ] [a[2, 1] a[2, 2]] > Determinant(A); a[1, 1] a[2, 2] - a[1, 2] a[2, 1] > A3 := Matrix(3,3): > for i from 1 to 3 do for j from 1 to 3 do A3[i,j]:=a[i,j]: od: od: > A3; [a[1, 1] a[1, 2] a[1, 3]] [ ] [a[2, 1] a[2, 2] a[2, 3]] [ ] [a[3, 1] a[3, 2] a[3, 3]] > Determinant(A3); a[1, 1] a[2, 2] a[3, 3] - a[1, 1] a[2, 3] a[3, 2] + a[2, 1] a[3, 2] a[1, 3] - a[2, 1] a[1, 2] a[3, 3] + a[3, 1] a[1, 2] a[2, 3] - a[3, 1] a[2, 2] a[1, 3] > A4 := Matrix(4,4); [0 0 0 0] [ ] [0 0 0 0] A4 := [ ] [0 0 0 0] [ ] [0 0 0 0] > for i from 1 to 4 do for j from 1 to 4 do A4[i,j]:=a[i,j]: od: od: > A4; [a[1, 1] a[1, 2] a[1, 3] a[1, 4]] [ ] [a[2, 1] a[2, 2] a[2, 3] a[2, 4]] [ ] [a[3, 1] a[3, 2] a[3, 3] a[3, 4]] [ ] [a[4, 1] a[4, 2] a[4, 3] a[4, 4]] > Determinant(A4); a[1, 1] a[2, 2] a[3, 3] a[4, 4] - a[1, 1] a[2, 2] a[3, 4] a[4, 3] + a[1, 1] a[3, 2] a[4, 3] a[2, 4] - a[1, 1] a[3, 2] a[2, 3] a[4, 4] + a[1, 1] a[4, 2] a[2, 3] a[3, 4] - a[1, 1] a[4, 2] a[3, 3] a[2, 4] - a[2, 1] a[1, 2] a[3, 3] a[4, 4] + a[2, 1] a[1, 2] a[3, 4] a[4, 3] - a[2, 1] a[3, 2] a[4, 3] a[1, 4] + a[2, 1] a[3, 2] a[1, 3] a[4, 4] - a[2, 1] a[4, 2] a[1, 3] a[3, 4] + a[2, 1] a[4, 2] a[3, 3] a[1, 4] + a[3, 1] a[1, 2] a[2, 3] a[4, 4] - a[3, 1] a[1, 2] a[4, 3] a[2, 4] + a[3, 1] a[2, 2] a[4, 3] a[1, 4] - a[3, 1] a[2, 2] a[1, 3] a[4, 4] + a[3, 1] a[4, 2] a[1, 3] a[2, 4] - a[3, 1] a[4, 2] a[2, 3] a[1, 4] - a[4, 1] a[1, 2] a[2, 3] a[3, 4] + a[4, 1] a[1, 2] a[3, 3] a[2, 4] - a[4, 1] a[2, 2] a[3, 3] a[1, 4] + a[4, 1] a[2, 2] a[1, 3] a[3, 4] - a[4, 1] a[3, 2] a[1, 3] a[2, 4] + a[4, 1] a[3, 2] a[2, 3] a[1, 4] > A5 := Matrix(5,5): > for i from 1 to 5 do for j from 1 to 5 do A5[i,j]:=a[i,j]: od: od: > A5; [a[1, 1] a[1, 2] a[1, 3] a[1, 4] a[1, 5]] [ ] [a[2, 1] a[2, 2] a[2, 3] a[2, 4] a[2, 5]] [ ] [a[3, 1] a[3, 2] a[3, 3] a[3, 4] a[3, 5]] [ ] [a[4, 1] a[4, 2] a[4, 3] a[4, 4] a[4, 5]] [ ] [a[5, 1] a[5, 2] a[5, 3] a[5, 4] a[5, 5]] > Delta[5]:=Determinant(A5); Delta[5] := -a[4, 1] a[2, 2] a[3, 3] a[1, 4] a[5, 5] + a[1, 1] a[5, 2] a[3, 3] a[2, 4] a[4, 5] + a[1, 1] a[5, 2] a[4, 3] a[3, 4] a[2, 5] - a[1, 1] a[5, 2] a[4, 3] a[2, 4] a[3, 5] - a[1, 1] a[5, 2] a[3, 3] a[4, 4] a[2, 5] + a[1, 1] a[5, 2] a[2, 3] a[4, 4] a[3, 5] - a[1, 1] a[4, 2] a[5, 3] a[3, 4] a[2, 5] - a[1, 1] a[5, 2] a[2, 3] a[3, 4] a[4, 5] + a[1, 1] a[4, 2] a[5, 3] a[2, 4] a[3, 5] - a[1, 1] a[4, 2] a[3, 3] a[2, 4] a[5, 5] + a[1, 1] a[4, 2] a[3, 3] a[5, 4] a[2, 5] + a[1, 1] a[4, 2] a[2, 3] a[3, 4] a[5, 5] - a[1, 1] a[4, 2] a[2, 3] a[5, 4] a[3, 5] - a[1, 1] a[3, 2] a[5, 3] a[2, 4] a[4, 5] + a[1, 1] a[3, 2] a[5, 3] a[4, 4] a[2, 5] - a[1, 1] a[3, 2] a[4, 3] a[5, 4] a[2, 5] + a[1, 1] a[3, 2] a[4, 3] a[2, 4] a[5, 5] + a[1, 1] a[3, 2] a[2, 3] a[4, 5] a[5, 4] - a[1, 1] a[2, 2] a[5, 3] a[4, 4] a[3, 5] - a[1, 1] a[3, 2] a[2, 3] a[4, 4] a[5, 5] - a[1, 1] a[2, 2] a[4, 3] a[3, 4] a[5, 5] + a[1, 1] a[2, 2] a[5, 3] a[3, 4] a[4, 5] + a[1, 1] a[2, 2] a[4, 3] a[5, 4] a[3, 5] - a[1, 1] a[2, 2] a[3, 3] a[4, 5] a[5, 4] + a[1, 1] a[2, 2] a[3, 3] a[4, 4] a[5, 5] - a[2, 1] a[4, 2] a[3, 3] a[5, 4] a[1, 5] + a[2, 1] a[4, 2] a[1, 3] a[5, 4] a[3, 5] - a[2, 1] a[4, 2] a[1, 3] a[3, 4] a[5, 5] + a[2, 1] a[3, 2] a[5, 3] a[1, 4] a[4, 5] - a[2, 1] a[3, 2] a[5, 3] a[4, 4] a[1, 5] + a[2, 1] a[3, 2] a[4, 3] a[5, 4] a[1, 5] - a[2, 1] a[3, 2] a[4, 3] a[1, 4] a[5, 5] + a[2, 1] a[3, 2] a[1, 3] a[4, 4] a[5, 5] - a[2, 1] a[3, 2] a[1, 3] a[4, 5] a[5, 4] + a[2, 1] a[1, 2] a[5, 3] a[4, 4] a[3, 5] - a[2, 1] a[1, 2] a[5, 3] a[3, 4] a[4, 5] + a[2, 1] a[1, 2] a[4, 3] a[3, 4] a[5, 5] - a[2, 1] a[1, 2] a[4, 3] a[5, 4] a[3, 5] - a[2, 1] a[1, 2] a[3, 3] a[4, 4] a[5, 5] + a[2, 1] a[1, 2] a[3, 3] a[4, 5] a[5, 4] + a[2, 1] a[4, 2] a[3, 3] a[1, 4] a[5, 5] - a[2, 1] a[4, 2] a[5, 3] a[1, 4] a[3, 5] + a[2, 1] a[5, 2] a[4, 3] a[1, 4] a[3, 5] - a[2, 1] a[5, 2] a[4, 3] a[3, 4] a[1, 5] + a[2, 1] a[5, 2] a[3, 3] a[4, 4] a[1, 5] - a[2, 1] a[5, 2] a[3, 3] a[1, 4] a[4, 5] + a[2, 1] a[5, 2] a[1, 3] a[3, 4] a[4, 5] - a[2, 1] a[5, 2] a[1, 3] a[4, 4] a[3, 5] + a[2, 1] a[4, 2] a[5, 3] a[3, 4] a[1, 5] - a[3, 1] a[1, 2] a[5, 3] a[4, 4] a[2, 5] - a[3, 1] a[2, 2] a[1, 3] a[4, 4] a[5, 5] + a[3, 1] a[1, 2] a[5, 3] a[2, 4] a[4, 5] + a[3, 1] a[1, 2] a[4, 3] a[5, 4] a[2, 5] - a[3, 1] a[1, 2] a[4, 3] a[2, 4] a[5, 5] + a[3, 1] a[1, 2] a[2, 3] a[4, 4] a[5, 5] - a[3, 1] a[1, 2] a[2, 3] a[4, 5] a[5, 4] - a[4, 1] a[1, 2] a[2, 3] a[3, 4] a[5, 5] - a[3, 1] a[5, 2] a[4, 3] a[1, 4] a[2, 5] + a[3, 1] a[5, 2] a[4, 3] a[2, 4] a[1, 5] - a[3, 1] a[5, 2] a[2, 3] a[4, 4] a[1, 5] + a[3, 1] a[5, 2] a[2, 3] a[1, 4] a[4, 5] + a[3, 1] a[5, 2] a[1, 3] a[4, 4] a[2, 5] - a[3, 1] a[4, 2] a[5, 3] a[2, 4] a[1, 5] - a[3, 1] a[5, 2] a[1, 3] a[2, 4] a[4, 5] - a[3, 1] a[4, 2] a[2, 3] a[1, 4] a[5, 5] + a[3, 1] a[4, 2] a[5, 3] a[1, 4] a[2, 5] - a[3, 1] a[4, 2] a[1, 3] a[5, 4] a[2, 5] + a[3, 1] a[4, 2] a[2, 3] a[5, 4] a[1, 5] + a[3, 1] a[4, 2] a[1, 3] a[2, 4] a[5, 5] + a[3, 1] a[2, 2] a[5, 3] a[4, 4] a[1, 5] + a[3, 1] a[2, 2] a[4, 3] a[1, 4] a[5, 5] - a[3, 1] a[2, 2] a[5, 3] a[1, 4] a[4, 5] + a[3, 1] a[2, 2] a[1, 3] a[4, 5] a[5, 4] + a[4, 1] a[3, 2] a[2, 3] a[1, 4] a[5, 5] - a[4, 1] a[3, 2] a[2, 3] a[5, 4] a[1, 5] - a[4, 1] a[3, 2] a[1, 3] a[2, 4] a[5, 5] + a[4, 1] a[3, 2] a[1, 3] a[5, 4] a[2, 5] + a[4, 1] a[2, 2] a[5, 3] a[1, 4] a[3, 5] - a[4, 1] a[2, 2] a[5, 3] a[3, 4] a[1, 5] - a[4, 1] a[2, 2] a[1, 3] a[5, 4] a[3, 5] + a[4, 1] a[2, 2] a[3, 3] a[5, 4] a[1, 5] + a[4, 1] a[1, 2] a[5, 3] a[3, 4] a[2, 5] + a[4, 1] a[2, 2] a[1, 3] a[3, 4] a[5, 5] - a[4, 1] a[1, 2] a[3, 3] a[5, 4] a[2, 5] + a[4, 1] a[1, 2] a[3, 3] a[2, 4] a[5, 5] + a[4, 1] a[1, 2] a[2, 3] a[5, 4] a[3, 5] + a[4, 1] a[5, 2] a[2, 3] a[3, 4] a[1, 5] - a[4, 1] a[5, 2] a[2, 3] a[1, 4] a[3, 5] + a[4, 1] a[5, 2] a[1, 3] a[2, 4] a[3, 5] - a[4, 1] a[5, 2] a[1, 3] a[3, 4] a[2, 5] - a[4, 1] a[3, 2] a[5, 3] a[1, 4] a[2, 5] + a[4, 1] a[3, 2] a[5, 3] a[2, 4] a[1, 5] + a[4, 1] a[5, 2] a[3, 3] a[1, 4] a[2, 5] - a[3, 1] a[2, 2] a[4, 3] a[5, 4] a[1, 5] - a[4, 1] a[1, 2] a[5, 3] a[2, 4] a[3, 5] - a[4, 1] a[5, 2] a[3, 3] a[2, 4] a[1, 5] + a[5, 1] a[1, 2] a[2, 3] a[3, 4] a[4, 5] - a[5, 1] a[1, 2] a[2, 3] a[4, 4] a[3, 5] - a[5, 1] a[2, 2] a[3, 3] a[4, 4] a[1, 5] + a[5, 1] a[2, 2] a[3, 3] a[1, 4] a[4, 5] - a[5, 1] a[2, 2] a[1, 3] a[3, 4] a[4, 5] + a[5, 1] a[2, 2] a[1, 3] a[4, 4] a[3, 5] + a[5, 1] a[1, 2] a[4, 3] a[2, 4] a[3, 5] - a[5, 1] a[1, 2] a[4, 3] a[3, 4] a[2, 5] + a[5, 1] a[1, 2] a[3, 3] a[4, 4] a[2, 5] - a[5, 1] a[1, 2] a[3, 3] a[2, 4] a[4, 5] + a[5, 1] a[3, 2] a[1, 3] a[2, 4] a[4, 5] - a[5, 1] a[3, 2] a[1, 3] a[4, 4] a[2, 5] + a[5, 1] a[2, 2] a[4, 3] a[3, 4] a[1, 5] - a[5, 1] a[2, 2] a[4, 3] a[1, 4] a[3, 5] - a[5, 1] a[4, 2] a[1, 3] a[2, 4] a[3, 5] + a[5, 1] a[4, 2] a[1, 3] a[3, 4] a[2, 5] + a[5, 1] a[3, 2] a[4, 3] a[1, 4] a[2, 5] - a[5, 1] a[3, 2] a[4, 3] a[2, 4] a[1, 5] + a[5, 1] a[3, 2] a[2, 3] a[4, 4] a[1, 5] - a[5, 1] a[3, 2] a[2, 3] a[1, 4] a[4, 5] - a[5, 1] a[4, 2] a[2, 3] a[3, 4] a[1, 5] + a[5, 1] a[4, 2] a[2, 3] a[1, 4] a[3, 5] - a[5, 1] a[4, 2] a[3, 3] a[1, 4] a[2, 5] + a[5, 1] a[4, 2] a[3, 3] a[2, 4] a[1, 5] > nops(Delta[5]); 120 >