Sage Project.sagews

#Matrices in Sage

#1.Define a Matrix A & Matrix B
A = matrix( [ [3, 6, 9], [7, 2, 4], [6, 8, 5] ])

print(A)

[3 6 9] [7 2 4] [6 8 5]

B = matrix( [ [1, 2, 3], [4, 5, 6], [7, 8, 9] ])

print(B)

[1 2 3] [4 5 6] [7 8 9]

#Notice with this method of writing it in Sage we do not need to define the size of the matrix.


#2. Simple Operations to the Matrix A & Matrix B.
#Multiplication
A*B

[ 90 108 126] [ 43 56 69] [ 73 92 111]

#Exponants
A^3

[1605 1602 1833] [1269 1238 1298] [1622 1696 1859]

A*A*A

[1605 1602 1833] [1269 1238 1298] [1622 1696 1859]

#3. Reducing a matrix
C= matrix([[3,2,1,4] , [2, 1, 5, 8] , [3,6,9,12]])

print C

[ 3 2 1 4] [ 2 1 5 8] [ 3 6 9 12]

print C.rref()

[ 1 0 0 6/5] [ 0 1 0 -2/5] [ 0 0 1 6/5]

#4. Inverse of a Matrix
C= matrix([[5,2,3] , [4, 9, 6] , [7,8,12]])

print"Before:"
print C

Before: [ 5 2 3] [ 4 9 6] [ 7 8 12]

print "After:"
C.inverse()

After: [ 4/13 0 -1/13] [ -2/65 1/5 -6/65] [-31/195 -2/15 37/195]

C^(-1)

[ 4/13 0 -1/13] [ -2/65 1/5 -6/65] [-31/195 -2/15 37/195]

#Using the Commands BEFORE and AFTER really help us be organized when typing our work.

#Instead of typing C.inverse(), we can use a different format.



#5. Determinants
D = matrix( [ [1, 2] , [3, 4]] )

print D

[1 2] [3 4]

det(D)

-2

#Determinants are useful , not only linear algebra, but also to do such things as finding out if a matrix is a group in Abstract Algebra.
#det= ad-bc, which in our case was det=(1*4)-(2*3)