HOMEWORK #1 due 9-16

NOTE: In the following, _ means subscript and ^ means superscript. The expression Sum(i=a,b) h(i) (for example) means the same as Sum_i=1^b h(i), which is the sum of h(i) for i=a,a+1,...,b. Also, ``ne'' means ``not equal to'', so that x ne 0 means that x is not equal to zero.

1. Consider the matrix

                (  -2    9   -7    3  -2  )
 A = (a_{ij}) = (   5   13   14    6   0  )
                (  11    0   17   -2  -3  )
 

2. Write down the three 3 by 3 matrices A=(a_{ij}) with entries
        (i) a_{ij}=i+j-2,
        (ii) a_{ij}=i^(j-1)    and
        (iii) a_{ij}=j-i
where i,j=1,2,3.

3. Let B be the matrix

         B = (  1  7 )
             (  4  3 )
             ( -3  6 )
 

4. Let A be the 3x3 matrix

         A = (  1   3   7 )
             (  2   6  14 )
             ( -1  -3  -7 )
 

5. Let A be an nxn matrix and j an nx1 column vector of 1s.
        (i) Show that j'A is a 1xn row vector whose elements are the column sums of A.
        (ii) Show that Aj is a nx1 column vector whose elements are the row sums of A.

6. Let A=xy' be the outer product of two 3x1 column vectors x and y, which is a 3x3 matrix since x is 3x1 and y' is 1x3. (Note that A_{ij}=x_iy_j for i,j=1,2,3.) In contrast, the inner product x'y = Sum(i=1,3) x_iy_i. Assume x and y are both nonzero. Show that
        (i)   If y'x=1, then A²=A
        (ii)   If y'x=0, then A²=0
        (iii)   tr(xy')=x'y.   (Warning: This is not a misprint:   xy' and x'y are different objects.)
        (iv)   rank(A)=1.