Determinant

Author

Eric W. Weisstein
April 19, 2024

This notebook downloaded from http://mathworld.wolfram.com/notebooks/LinearAlgebra/Determinant.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Determinant.html.

©2024 Wolfram Research, Inc. except for portions noted otherwise

3x3

det=Det[{{a1,a2,a3},{b1,b2,b3},{c1,c2,c3}}]

{
-(a1 Det[(({{c2, b2}, {c3, b3}}))]+b1 Det[(({{a2, c2}, {a3, c3}}))]-c1 Det[(({{a2, b2}, {a3, b3}}))]),
-Det[(({{a1, c1, b1}, {a2, c2, b2}, {a3, c3, b3}}))]
det
}//Equal

{
-(a1 Det[(({{c2, b2}, {c3, b3}}))]-b1 Det[(({{c2, a2}, {c3, a3}}))]+c1 Det[(({{b2, a2}, {b3, a3}}))]),
-Det[(({{c1, b1, a1}, {c2, b2, a2}, {c3, b3, a3}}))]
det
}//Equal

Cayley-Menger determinant

d1=Det[{{0,1,1,1},{1,0,c^2,b^2},{1,c^2,0,a^2},{1,b^2,a^2,0}}];
d2=Det[{{0,a,b,c},{a,0,c,b},{b,c,0,a},{c,b,a,0}}];

{d1-d2,d1}

A055546

Table[(-1)^(j+1)2^j j!^2,{j,0,5}]

Determinants by inversions

3x3

Permutations[Range[3]]

Signature/@%

Inversions/@%

Det[{{(Subscript[a, 1]),(Subscript[a, 2]),(Subscript[a, 3])},{(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3])},{(Subscript[c, 1]),(Subscript[c, 2]),(Subscript[c, 3])}}]

Plus@@((-1)^Inversions[#]Times@@Thread[Subscript[{a,b,c},#]]&/@Permutations[Range[3]])

%-%%

4x4

Det[{{(Subscript[a, 1]),(Subscript[a, 2]),(Subscript[a, 3]),(Subscript[a, 4])},{(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3]),(Subscript[b, 4])},{(Subscript[c, 1]),(Subscript[c, 2]),(Subscript[c, 3]),(Subscript[c, 4])},{(Subscript[d, 1]),(Subscript[d, 2]),(Subscript[d, 3]),(Subscript[d, 4])}}]

Plus@@((-1)^Inversions[#]Times@@Thread[Subscript[{a,b,c,d},#]]&/@Permutations[Range[4]])

%-%%

Array[a,4]

Hadamard's maximum determinant theorem

Table[n^(n/2),{n,10}]

N[%]

(n^(n/2))^2

Table[n^n,{n,10}]

CircularDistribution[num_]:=Table[Sqrt[RandomReal[]]Exp[2π I RandomReal[]],num]

data=Table[Abs[Det/@Table[CircularDistribution[n],10^6,n]],{n,2,4}];//Timing

GraphicsRow[Table[Histogram[data[[n-1]],100,ChartStyle->Red,Frame->True,PlotLabel->Row[{n,"×",n}],FrameLabel->{Abs[det[A]],Style["count",Italic]},LabelStyle->Directive[12,FontFamily->"Times"]],{n,2,4}],ImageSize->800]