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(-4..4));

Using B1 defined as follows
> B1:=Copy(B):B1[1,2]:=x:B1[3,3]:=y:B1;
find the determinant of B1, and use 
> s:=solve(Determinant(B1)=0,x); 
to find the value of y which makes B non-singular and check
your result using the commands Determinant and subs.

Repeat the above by instead setting B[3,2] to be y 
and compare your two results. 

Using 2 random 2x2 matrices P and Q, check that the determinants of P and Q 
when multiplied give the same as the determinant of PQ and QP and
that the determinants of the transpose and the inverse of P also give the 
expected answer.

Predict and then calculate the eigenvalues and eigenvectors of this matrix:
> G:=Matrix(<<4|0|0|1|0>,<0|3|0|2|1>,<0|0|-1|1|1>,<0|0|0|2|3>,<0|0|0|0|-3>>);
Use IdentityMatrix and then DeleteRow and DeleteColumn to achieve the Lapace 
Expansion as last week and check your answer with 
>   Eigenvalues(G);
and 
>   (v,e):=Eigenvectors(G);
reading carefully the help pages which describe their format

Check that 
> e2:=e[1 .. -1, 2];
extracts the 2nd eigenvector and verify that 
> Multiply(G, e2);
is the same as
> v[2]*e2;
for all of the five eigenvectors.

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>>);
only two eigenvectors, using 
> Eigenvalues(J);
and then solving the two resulting homogeneous systems of equations.
Verify your answer using 
> Eigenvectors(J);

Using Eigenvectors(K) see that 
> K:=Matrix([[5, 5/2, 1], [-2, -1, -2], [-2, -5, 2]]);
has two different eigenvalues but three different eigenvectors.
Check that although 
> v4:=<7,-4,3>;
(hopefully) does not appear as one of your eigenvectors it still satisfies 
the eigenvector equation somehow. Find a v5 that also has this property.

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

Use determinant row and column operations to make smaller numbers
gradually appear and hence find the remaining eigenvalues of L, using what 
you have already learned to help you isolate the factors you need.

What are the eigenvalues of H^5 ? 
Try to predict the eigenvalues and eigenvectors of J^3 and K^4 
and check your answer.