# matrix(Matrix) -- the matrix between generators

## Synopsis

• Function: matrix
• Usage:
matrix f
• Inputs:
• f, , a map of modules
• Optional inputs:
• Degree => ..., default value null, create a matrix from a doubly-nested list of ring elements or matrices
• Outputs:
• , If the source and target of f are free, then the result is f itself. Otherwise, the source and target will be replaced by the free modules whose basis elements correspond to the generators of the modules.

## Description

Each homomorphism of modules $f : M \rightarrow N$ in Macaulay2 is induced from a matrix $f0 : \mathtt{cover} M \rightarrow \mathtt{cover} N$. This function returns this matrix.
 i1 : R = QQ[a..d]; i2 : I = ideal(a^2,b^2,c*d) 2 2 o2 = ideal (a , b , c*d) o2 : Ideal of R i3 : f = basis(3,I) o3 = {2} | a b c d 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 a b c d 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 a b c d | o3 : Matrix i4 : source f 12 o4 = R o4 : R-module, free, degrees {12:3} i5 : target f o5 = image | a2 b2 cd | 1 o5 : R-module, submodule of R
The map f is induced by the following 3 by 12 matrix from R^12 to the 3 generators of I.
 i6 : matrix f o6 = {2} | a b c d 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 a b c d 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 a b c d | 3 12 o6 : Matrix R <--- R
To obtain the map that is the composite of this with the inclusion of I onto R, use super(Matrix).
 i7 : super f o7 = | a3 a2b a2c a2d ab2 b3 b2c b2d acd bcd c2d cd2 | 1 12 o7 : Matrix R <--- R