Math 427K-H   Homework 6    Due Thursday       March 12






1.  The notation in matlab for a  matrix is A = [ 1  2; 1  3;  2  4] 

where  the  semicolons separate  rows. The matrix A above is  3x2.

A  vector is written v = [1; 3 ; 5]  and a covector or row vector is

is w = [ 1  2   5].  Suppose B = [1 2 4; 2 4 0].


Compute for the above A  and B the following:

wA, Bv,  wv, vw , (v')v ,  w w'  , A' A  and A A'.  Here we 

are using A' to indicate the complex (Hermitian) transpose of A, because

that is what matlab uses.  the transpose of A = A' in matlab when A is real.




2. Let  A =  [ 1  0  0; 1  2  0 ;  2  3  4]  and B =  [1 4; 2 9].  Compute

A^-1 and B^-1 in any way you like. 



3.  Let  y ' = A y  where A = [3 -2; 2, -2].  Here y = [ y(1); y(2)] is a

column vector and A is a 2X2 matrix.   Find r so that y(t) =

[4 ; 2] exp(rt) solves the vector differential equation.


4. Use a computer program (matlab is available and I can help you with it)


to compute the eigenvalues, eigenvectors  and the determinant of the matrices


A = [3 , 2 ; 2, -2]   and B = [1  4 ; 2  9].

5. Find a basis for the solution space of vectors  v = 
[v(1); v(2 ;  v(3) ; v(4)]  of the equation Av = 0,  where A is the
matrix [1 1 1 1; 1 2 3 4.]. 


Extra Credit:  Suppose that N is a strictly upper triangular nxn matrix.

Show that N^n = 0.  Now show that (I - N )^-1 =  I  +  N + N^2+ ....+  N^(n-1).

Suggestion:  work it out for 3X3 matrices first (half credit).

Note that you don't have to compute the inverse.  It is only necessary

to show  that the given matrix works as an inverse.