# MAPLE ASSIGNMENT Matrix part 1
# The first command loads the matrix algebra operations into Maple.
# Change the : to ; if you wish to see all the commands in this package.
> with(linalg):
# Next define a few matrices. You can change the numbers to any numbers
# you like.
> M1:=matrix(3,3,[1,0,-1,1,5,7,3,3,9]);
> M2:=matrix(3,3,[1,0,-1,3,6,21,3,3,9]);
# The next commands create some other matrices from M1 and M2.
> M3:=submatrix(M2,2..3,1..2);
> M4:=submatrix(M1,2..3,1..3);
> ### WARNING: linalg[stack] has been renamed linalg[stackmatrix] to
> avoid collision with mainline stack
> M5:=stack(M2,M4);
> M6:=concat(M3,M4);
#  The next command illustrates multiplication by a scalar.  Evalm is
# needed for the answer to appear 
# on the screen. Refer back to M6 to see what has happened
> evalm(3*M6);
# 
# The aritmetic operations for matrices are + , - , &* , ^ . For easier
# reference the next command 
# brings M1 and M2 back on to the screen side by side in a 3X6 matrix.
# Execute the next 
# command and after you are sure you understand what has happened
# execute the command agaiin 
# with + changed first to - and then to &* .
> concat(M1,M2);
> evalm(M1+M2);
# If you wish to check your understanding of matrix multiplication the
# following command calculates the 
# [2,2] entry of the product matrix. Change the numbers in this command
# to calculate some other 
# entries in the product matrix. If one of the numbers is negative put
# it in brackets.
> (1*0)+(5*6)+(7*3);
# The next comand does a power of a matrix. You can check the answers
# out in the same way the 
# product was checked.
> evalm(M1^2);
# The following illustrate some of the properties of matrix algebra. A
# matrix is transposed when its 
# rows and columns are switched..  The properties are the associative ,
# commutative and distributive 
# properties.
> M7:=transpose(M4)&*M4;
> evalm(M4);
> evalm(M7);
> evalm((M1+M2)+M7);
> evalm(M1+(M2+M7));
> evalm((M1&*M2)&*M7);
> evalm(M1&*(M2&*M7));
> evalm((M1+M2)-(M2+M1));
# The commutative law does not however work with multiplication.
> evalm((M1&*M2)-(M2&*M1));
# The final property illustrated here is one of the distributive laws.
> evalm((M1+M2)&*M7);
> evalm((M1&*M7)+(M2&*M7));
# Diagonal matrices are created with the diag command.  This can be used
# to create an identity matrix.
> diag(-3,2,Pi); 
> iden:=diag(1,1,1);
# Inverse matrices are either A^(-1)  or inverse(A).  The following
# calculate the inverse, check that it 
# works and illustrate the properties of inverses.  You should also try
# to calculate the inverse of M2 .
> M8:=M1^(-1);
> evalm(%);
> evalm(M1&*M8-iden);
> evalm(M8&*M1-iden);

> evalm(M8^(-1));
> M9:=evalm(M1+M7);
# The next commands illustrate the correct and the incorrect way to get
# the inverse of the product of twp matrices.
> evalm((M1&*M9)^(-1));
> evalm((M9^(-1))&*(M1^(-1)));
> evalm((M1^(-1))*(M9^(-1)));
> evalm((M1^(-1))+(M9^(-1)));
> evalm((M1+M9)^(-1));
> 
# The last two commands have illustrated a dumb artihmetical mistake you
# should not make even with numbers.