As last week, start Maple and load the linear algebra progams using:
>    with(LinearAlgebra);

This week we will be using our row operations to see matrices which
are consistent and inconsistent (have and do not have solutions)

As you learned to do RowOperation last week, you will be allowed to use Pivot this week.
To apply Pivot to a column you should identify a 1 in row r and column c in Matrix M.
Applying the command Pivot(M,r,c);
will select and apply the row operations to make the zeros above and below your 1 in column c.
(Note that Pivot will work with any non-zero number, but I only want you to apply it to a 1 this week)

Let us consider this system of equations where the top line represents the equation 4*w+x+2*y+z = 9, etc.

Q1: Use a sequence of Pivot and RowOperation commands to show that this system of equations is inconsistent (has no solution) unless p is a particular integer. Identify the value of p that makes the equations consistent.
>  B0:=Matrix(4, 5, [[4, 1, 2, 1, 9], [-2, 7, -4, 1, 3], [-3, 6, 1, 3, -4], [1, 5, -4, 0, p]]);
[2 marks]

Q2:
 Let
> C0:=Matrix(4, 5, [[-1, 1, 2, 1, 5], [2, -1, -1, 3, -4], [3, -1, 0, 7, -3], [4, -1, 1, 11, -2]]);
use 
> ReducedRowEchelonForm(C0);
to see the RREF of this system of equations.
Using paper identify a particular and two different homogeneous solutions corresponding 
to this form and let them be w, x, y and z, and then check using:

> -w + x + 2*y + z; 2*w - x - y + 3*z; 3*w - x + 7*z; 4*w - x + y + 11*z;

Now explain using # why w:=8; x:=13; y:=1; z:=-2; is also a particular solution
and why w:=5; x:=-6; y:=7; z:=-3; is also a homogeneous solution
[2 marks]

Define these two matrices:
> E:=<<3,1,-2>|<1,3,0>>;
> F:=<<1,2>|<-1,3>>;

Q3
Note what Maple's pink error response is when you try to add or multiply them using:
> G1:=E+F;
> G2:=Multiply(F,E);
Explain using # why these results are as you would expect.
[1 mark]

Q4
You can create the matrix we use for finding the inverse as follows:
> H0:=<F|IdentityMatrix(2)>;
Use RowOperation or Pivot to create the inverse as described in class.
Check that on the right of the imaginary dotted line is the same as
> MatrixInverse(F);
[1 mark]
(Note that Maple will give an inverse for a non-square matrix, 
you should ignore that until you get to take Linear Algebra 2)

Q5
Create a random 3x3 matrix M with rank 3 as follows:
> randomize(Value(Time())): M:=<1>: while Rank(M)<3 do M:=RandomMatrix(3,3,generator=rand(-4..4)); od: M;
We can check the determinant of M (and that it is non-zero) using:
> Determinant(M);

Create 
> M0:=<M|IdentityMatrix(3)>;
Make a 1 if necessary and Pivot around it, then make another 1 and Pivot around it. 
The final number left to Pivot should be either plus or minus the determinant

Complete your Pivot after multiplying this row by a number to make the pivot point 1.

Use the Row swapping RowOperation to make the left matrix exactly like the Identity
and compare your answer with Maple:
> MatrixInverse(M);
[3 marks]

Q6
Create two 2x2 matrices of rank 2 which when added produce a matrix of rank 1.
Check their Rank using Maple and try to pick ones that nobody else in the class will have chosen.
[1 mark]


Once you have completed all of these activities, please save your worksheet then use
File / Export As and choose pdf. Now upload the file to Crowdmark.