Math 427K-H   Assignment 7  Due  March  24





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.


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

A^-1 and B^-1 in any way you like, but I expect you to use the computer for

the 3X3 matrix. 



2.  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.


3. Use a computer program (matlab is available on the math computers)

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

A = [3 , 2 ; 2, -2],  B = [1  4 ; 2  9] and C =[1, 3, 1 ; 1,0,4 ; 0, 1, 2].


4. 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.].

5.  


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.