Start Maple from Start Menu/Program Files or using the icon
in f:\math115

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

Create a random 5x4 matrix with entries between -4 and 3 using
>   P:=randmatrix(5,4,entries=rand(-4..3));

Use the LU algorithm to produce U and deduce L from the process.
Check that multiply(L,U) is equal to P.

Find the inverse of L and verify that it is also lower triangular
What about the inverse of U?

What is the LU factorisation of Q:=diag(4,1/2,-3); ? 


Create a general 2x2 matrix X with the following command
>   X:=matrix(2,2,[a,b,c,d]);
and check that the inverse is as the usual formula given states.

Use the command 
>   det(X);
to get the so called determinant of the matrix.

Create a 3x3 random matrix M and check that the rank is equal to 3

Use determinant row and column operations (addrow, addcol) on M 
to check your answer, checking the determinant is unchanged after each operation.

Verify that swapping any two rows or columns at any point changes the sign of the determinant

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 determinant of the new matrix M.

Use row ops and swaps 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:evalm(M);
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 scalar 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.