Written Assignment 1
> restart;
> with(LinearAlgebra):
In each part, use maple to perform the following steps. (Problem 1 is a bit different.)
1 Find the characteristic polynomial of the given matrix and find the eigenvalues
of .
2 For each eigenvalue, find a basis of the eigenspace for that eigenvalue (use the
N u l l S p a c e command).
3 Determine if
that
4 If
is diagonalizable. If so find a diagonal matrix
Use Maple to check your answer.
and a matrix
so
is not diagonalizable, you can stop here, otherwise go on.
5 Find the Matrices
and
.
6 For the given initial condition , find the solution of the intial value problem
,
.
Problem 1
In this problem, carry out the steps above using hand computation. Show your
work. You may work on different paper and attach it to the assignment.
> A := Matrix(2, 2, {(1, 1) = 0, (1, 2) = -2, (2, 1) = 1, (2, 2)
= 3}); c := Vector(2, {(1) = -1, (2) = 1});
Problem 2
> A := Matrix(2, 2, {(1, 1) = -18, (1, 2) = -20, (2, 1) = 15, (2,
2) = 17}); c := <-1,2>;
Problem 3
> A := Matrix(3, 3, {(1, 1) = -98, (1, 2) = 25, (1, 3) = -5, (2,
1) = -956, (2, 2) = 241, (2, 3) = -48, (3, 1) = -2880, (3, 2) =
720, (3, 3) = -143});
> c := Vector(3, {(1) = -1, (2) = -2, (3) = 3});
Problem 4
> A := Matrix(3, 3, {(1, 1) = -4441, (1, 2) = -30192, (1, 3) =
9990, (2, 1) = -240, (2, 2) = -1633, (2, 3) = 540, (3, 1) =
-2700, (3, 2) = -18360, (3, 3) = 6074});
> c := Vector(3, {(1) = 1, (2) = -5, (3) = 2});
Problem 5
> A := Matrix(3, 3, {(1, 1) = -4297, (1, 2) = -29211, (1, 3) =
9666, (2, 1) = -240, (2, 2) = -1633, (2, 3) = 540, (3, 1) =
-2636, (3, 2) = -17924, (3, 3) = 5930});
> c := Vector(3, {(1) = 1, (2) = 1, (3) = 1});
Problem 6
> A := Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 15, (2, 1)
= 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 1, (3, 3) =
1});
> ic := Vector(3, {(1) = 1, (2) = 1, (3) = 1});