Problem Set 4

1. Given:

            __         __
            |1    0    3|
        A = |           |
            |2   -1    1|
   
   
            __         __                 __ __ 
            |3    4    1|                 |  2|      
            |           |                 |   |                             
        B = |0   -1    1|             C = | -1|
            |           |                 |   |
            |1    2   -2|                 |  4|
   
   
   

   Calculate by hand (check with Stata)
   1. (AB)'
   2. B'A'
   3. (AC)'
   4. C'A'

    

2. Given:

         __       __                        __   __
         |4   3   1|                        |2   1|
         |         |                        |     |
     A = |2   2   5|                    C = |3   0|
         |         |                        |     |
         |0   3   1|                        |5   5|   
 
                        b' = [ 1  1  2 ] 

Calculate by hand (except for e) and check with Stata the following:
1. AC
2. Ab
3. A'
4. A'A
5. A^-1 (use Stata)
6. AA^-1 (don't do any matrix manipulations)




3. Write out the following equations in matrix form:

   9X₁ + 2X₂ + 4X₃ + X₄ = 5
   2X₁ + X₄ = 6
   3X₂ - X₃ = 1
   10X₁ + 4X₃ + 8X₄ = 12

   Using Stata, solve for X₁, X₂, X₃, and X₄.





4. Do the following matrix multiplications:
   

1.          __          __   __ __
            | 1  0  1  0 |   | a |
            | 0  1  0  1 |   | b |
                             | c |
                             | d | 

2.          __          __   __ __
            | 1 -1  0  0 |   | a |
            | 0  0  1 -1 |   | b |
                             | c |
                             | d | 

3.                           __ __ 
            [ 1  1  1  1 ]   | a |
                             | b |
                             | c |
                             | d |

4.          __          __   __ __
            | 1  0  0  0 |   | a |
            | 0  1  0  0 |   | b |
            | 0  0  1  0 |   | c |
            | 0  0  0  1 |   | d |

5.                           __ __ 
            [ 0  1  0  0 ]   | a |
                             | b |
                             | c |
                             | d |

Express your results in terms of the symbols a, b, c, and d.





5. Define the following:
   1. The sum of the matrices D (m x n) and E (m x n)
   2. The product of the matrices P (m x n) and Q (n x o)
   3. A non-singular matrix.
   4. The rank of a matrix
   5. A vector
   6. The transpose of a matrix
   7. The identity matrix; the null matrix.
   8. The inverse of a matrix; when does the inverse of a matrix not exist?

Bert Kritzer, 608-263-2277, Kritzer@PoliSci.Wisc.Edu
Last modified, February 26, 2004