#Matrices in Sage

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

print(A)

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

print(B)

#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

#Exponants
A^3

A*A*A

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

print C

print C.rref()

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

print"Before:"
print C

print "After:"
C.inverse()

C^(-1)

#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

det(D)

#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)