by its entries
Using the function
matrix is the most basic method for inputting a matrix. The entries are typed in by rows.
i1 : R = ZZ/5[s..z];

i2 : M = matrix {{ x^2+y, z^3}, {y^3z,3*z6*x5*y}}
o2 =  x2+y z3 
 y3z x2z 
2 2
o2 : Matrix R < R

by a function
The function
map can be used to construct matrices.
i3 : G = map(R^3,3,(i,j)>R_i^j)
o3 =  1 s s2 
 1 t t2 
 1 u u2 
3 3
o3 : Matrix R < R

i4 : f = 3*s^2*vt*u*v+s*t^2
2 2
o4 = s*t  2s v  t*u*v
o4 : R

i5 : H = map(R^4,R^4,(i,j)>diff(R_j*R_i,f))
o5 =  v 2t 0 s 
 2t 2s v u 
 0 v 0 t 
 s u t 0 
4 4
o5 : Matrix R < R

identity matrix
The function
id is used to form the identity matrix as a map from a module to itself.
i6 : id_(R^3)
o6 =  1 0 0 
 0 1 0 
 0 0 1 
3 3
o6 : Matrix R < R

i7 : id_(source M)
o7 = {3}  1 0 
{3}  0 1 
2 2
o7 : Matrix R < R

The first example gives a 3x3 identity matrix with entries in the ring R. The second gives a 2x2 identity matrix whose source and target are the (graded) source of the matrix
M.