Start Maple from Start Menu/Program Files or using the icon
in f:\math115, or double click on 115lab3base.mws

Load the linear algebra progams with the command:
>    with(linalg):

Create a random 4x4 with entries between 1 and 5 using
>   M:=randmatrix(4,4,entries=rand(1..5));
Use the command

>   M[2,3]:=x:evalm(M)
to assign a new (unknown) value to one particular entry of your
matrix and display it. 

Use
>   det(M);
to find the new determinant.

Use determinant row and column operations (addrow, addcol) on M 
to check your answer, checking the determinant after each operation.
Use swaprow and swapcol to reduce your matrix to a triangular one and
verify that the determinant is just the product of the diagonal entries.

You can use
>   singu:=solve(det(M)=0,x);     # or just solve(det(M));
to find what value of x gives a non-invertible matrix.

What does Maple say the inverse of M is?
Change M to be the matrix with the nasty value by using
>   M[2,3]:=singu;
and check that the rank of M has decreased and that det(M) is zero.
Check that inverse(M) does not work any more too.


Create several random 3x3 matrices A, B and C and find their
determinants. Check the values of det(AB), det(inverse(C)) and det(ABC)
and relate them to the determinants of A, B and C.


Verify that multiplying any row of your matrix B by a number k changes the 
determinant by a factor of k, and that det(k*B) matches the expected formula.

Verify that det(A+C) is not equal to det(A) + det(C) (unless you were really unlucky)


Using
>  E := matrix([[-8, -5, 3], [22, 13, -6], [16, 8, -1]]);

form the matrix 
>  E1:=matadd(E,diag(lambda,lambda,lambda),1,-1);

Use determinant row and column operations on E1 to find which values of lambda
give a singular matrix.

Check your answer with 
>  det(E1);
and
> factor(det(E1));

and then
> eigenvals(E);
and
> eigenvects(E);

multiply E by the vectors in the last command (using them as columns) and look
for a relation between the answers.