Minor

Author

Eric W. Weisstein
February 10, 2003

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

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

Code

Minor[m_List?MatrixQ,{i_Integer,j_Integer}]:=Det[Drop[Transpose[Drop[Transpose[m],{j}]],{i}]]

MinorMatrix[m_List?MatrixQ]:=Map[Reverse,Minors[m],{0,1}]

Minor is equivalent to

MinorMatrix[[i,j]]

Usage

?Minors

©2005 Wolfram Research, Inc. except for portions noted otherwise

Examples

Analytic

?InvisibleSpace

Minors[Array[a,{5,5}],1]/.a[x_,y_]:>Subscript[a,ToString[x]<>ToString[y]]//MatrixForm//TraditionalForm

(({{(Subscript[a, "11"]), (Subscript[a, "12"]), (Subscript[a, "13"]), ⋯, (Subscript[a, "1\!\(\* StyleBox[\"n\",\nFontSlant->\"Italic\"]\)"])}, {(Subscript[a, "21"]), (Subscript[a, "22"]), (Subscript[a, "23"]), ⋯, (Subscript[a, "2\!\(\* StyleBox[\"n\",\nFontSlant->\"Italic\"]\)"])}, {(Subscript[a, "31"]), (Subscript[a, "32"]), (Subscript[a, "33"]), ⋯, (Subscript[a, "3\!\(\* StyleBox[\"n\",\nFontSlant->\"Italic\"]\)"])}, {⋮, ⋮, ⋮, ⋱, ⋮}, {(Subscript[a, k1"]), (Subscript[a, "\!\(\* StyleBox[\(\* StyleBox[\"k\",\nFontSlant->\"Italic\"]2\)]\)"]), (Subscript[a, k3"]), ⋯, (Subscript[a, "\!\(\* StyleBox[\"kn\",\nFontSlant->\"Italic\"]\)"])}}))

Minors[Array[a,{4,4}],2]/.a[x__]:>Subscript[a,x]//MatrixForm

Minors[Array[a,{4,4}],3]/.a[x__]:>Subscript[a,x]//MatrixForm

Minors[Array[a,{4,4}],4]/.a[x__]:>Subscript[a,x]//MatrixForm

Minors[Array[a,{4,4}]]/.a[x__]:>Subscript[a,x]//MatrixForm

Numerical

MatrixForm[m=Table[Random[Integer,{1,5}],{3},{3}]]

Minors[m]

Minors[m]

MinorMatrix[m]

Table[MinorMatrix[m][[i,j]],{i,3},{j,3}]

Table[Minor[m,{i,j}],{i,3},{j,3}]

Table[Minors[m][[j,i]],{i,3},{j,3}]