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#        Math 2270 Sec. 2                           MAPLE Assignment  #1.

        

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#Here are some commands which you might need for this  project.

>with(linalg):                    # Loads the linear algebra routines.
>with(plots): #Loads packages to plot graphs of functions
>A:=matrix([[1,2,6],[3,4,5],[7,9,11]]);
               [  1  2  6  ]
        A:=    [  3  4  5  ]
               [  7  9  11 ]
>A[2,3];                          # Gives the entry for row  2  and column  3.
                     5
>rref(A); # Computes the reduced row eachlon form of A.
>det(A);                          # Computes the determinant.
                    -3
>transpose(A);                    # computes the transpose
               [  1  3  7  ]
               [  2  4  9  ]
               [  6  5  11 ]
>det(%);                          # The % sign  is shorthand for the 
#                                   immediately preceding display. 
#                                   (determinant of A in this case.)
                    -3

># Here are some more instructions

>band(V,n);                       # Makes an  n x n  banded matrix from the 
#                                   values given for the left and top by the
#                                   2n-1  dimensional vector  V. Try 
#                                   >band([0,1,2,3,4,5,6,7,8,9,10],6);
>inverse(A);                      # computes the inverse. Same as >evalm(A^(-1));
>evalm(A+B);                      # Adds two matrices.
>multiply(A,B);                   # Multiplies two matrices. Same as 
#                                   >evalm(A &* B);
>minor(A,i,j);                    # Gives the  (n-1) x (n-1) submatrix obtained 
#                                   by removing the i-th row and j-th column
>adj(A);                          # Computes the transpose of the cofactor matrix.
>mulrow(A,i,c);                   # Creates a new matrix by multiplying the  i-th 
#                                   row of  A  by  the expression  c.
>addrow(A,i,j,c);                 # Creates a new matrix by replacing row j of 
#                                   matrix  A  with the row formed by   
#                                    c * row i + row j.
>swaprow(A,i,j);                  # Exchanges row i  and  row j  of matrix  A.
 

#PROBLEMS:


#     (This project was inspired by problems of Multivariable Mathematics with MAPLE,         
#         by J. Carlson  &  J. Johnson, p.27)

#You are to create a document in which you answer the following questions, using a mixture 
#of MAPLE computations and textual insertions (using # to comment or handwritten text.)
#Print out a copy and hand it in. Remember to put your name and section number on your 
#paper.

#Define   
#                (  2  3  4  )                 (  3  2  1  )
#  A    =        (  5  6  7  ) ,       B  =    (  4  3  2  ) .
#                (  8  9  0  )                 (  5  4  3  )

# 1.a     Compute  AB  and  BA.  Are they the same?
# .b     Compute  A + B  and  B + A.   Are they the same?
# .c     Define   C   to be   A + B.   Compute C^2   and compare it to  
#                        A^2 + 2AB + B^2.        
#        Are they the same?  Can you think of a small change you could make in the       
#        expression A^2 + 2AB + B^2 in order to make it equal to C^2 ?  
# .d     Compute the transpose of  AB  and compare it to the product of the transpose of  A
#        and the transpose of  B  multiplied in the correct order to get equality. 
# .e     Define  v = (1,2,3) to be a vector. Compute  Av.  
#        What does MAPLE  give you when you try  vA?
# .f     Solve  Ax = v  in three ways where  v  is the vector in  (1.e): by row reducing 
#        the augmented matrix, by using the command   linsolve   , and by using the inverse 
#        matrix of   A. 
#
#2.a     Solve  Bx = w  where  B is  as above  and  w=(1,2,3). Verify that your       
#        solution solves  Bx = w.
# .b     Repeat your work in order to solve  Bx = z  where  z = (1,2,4). Explain        
#        your answer.
#
#3.a     Plot the graph of the line x/2 + 3y=7  on the interval x=2..4 
#	using both plot and implicitplot commands 
#	
# .b     By plotting lines (anyway you like) determine if the system of 
# 	equations
#	
#	3x+2y=5, 2x-y=3
#	
#	has a solution and if the solution is unique