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

As usual, start Maple and load the linear algebra functions. 

Using the command "Determinant" find the determinant of the random 3x3 
matrix B and verify that it doesn't change at each step when you do 
determinant row and column operations to reduce it to row echelon form.
> B:=RandomMatrix(3,3,generator=rand(-3..3));

Using
> B[1,2]:=x;
set one entry of your matrix to be the unknown x and find the determinant
of this new matrix, using 
> s:=solve(Determinant(B)=0); 
to find the value of x which makes B singular.
Verify your result using
> Determinant(subs(x=s,B));

Repeat the above setting B[3,2] to be y and then B[3,3] to be y instead
and compare your two results. Find the values of x and y which give 
a matrix which is singular or never singular in each case.

Using 3 random 2x2 matrices P, Q and R, check for a relation between
the determinants of P, Q and R and those of PQ, QR, PR and PQR, and 
note that there is no similarly nice relation between those of P+Q or P-Q

Predict and then calculate the eigenvalues and eigenvectors of this matrix:
>G:=Matrix(<<4|5|1|-2|0>,<0|3|-2|2|1>,<0|0|7|1|-2>,<0|0|0|4|-3>,<0|0|0|0|-3>>);

Check your answer with 
>   Eigenvalues(G);
and 
>   (v,e):=Eigenvectors(G);
reading carefully the help pages which describe their format

Check that 
> Multiply(G, e[1 .. -1, j]);
is the same as
> v[j]*e[1 .. -1, j]
for j from 1 to 5


Given H:=Matrix([[-1, 3, -3], [5, -3, 6], [6, -6, 9]]);
check that although H has rank 2 it has 3 different eigenvalues
by using determinant operations (including ColumnOperation) on 
> H1 := H-lambda*IdentityMatrix(3)
Check your answer using 
> factor(Determinant(H1));

Show that
>  J:=Matrix(<<-2, 6, 2>| <-2, 5, 1>| <-2, 4, 2>>);
has rank 3 but only two eigenvectors, again using determinant operations
and verifying your answer using 
> (v, e) := Eigenvectors(J);

Given this matrix
>   K:=Transpose(Matrix([[115, 198, 87, 81], [-126, -218, -96, -90], [72, 126, 55, 54], [66, 114, 51, 46]]));
Verify by multiplication that 
> v1:=Vector(<-5, -4, 0, 1>);
and 
> v2:=Vector(<2, 3, 2, 0>);
are eigenvectors and deduce their eigenvalues.

Use determinant row and column operations to find the remaining 
eigenvalues of K, using what you have already learned to help you 
isolate the factors you need.


What are the eigenvalues of H^6 ? 
Try to predict the eigenvalues and eigenvectors of J^6 and K^6 
and check your answer.

Is there any pattern to the eigenvalues of H+J ? HJ ? JH ?