12.7 Solutions to First Review

This review is quite long, but it provides a thorough review of the information covered. You might want to give the students a few days to work on this assignment. If they can work all these problems, then the test should not be difficult.

1. 2 by 3
   

2.  
   

3. 1.  
      

   2.  
   
   

4. 1.  
      

   2.  
   
   

5. Yes because B = B^T
   

6. 1. 
      

   2. Freshmen
      

   3.  
      

   4. 2900
      

   5.  

      These solutions are given in dollars because we represented 30 cents as 0.3.
      

   6. $1540
   
   

7. 1. No because the inside dimensions of 2 by 3 multiplied by 2 by 3 do not match.
      

   2.  

      Remember that the students were only asked for 2 of these.

   
   

8.  
   

9.  
   
   
   

   
   

10.  
    
    

    
    

11.  
    
    

    
    

12. No. If the matrix is inconsistent or underdetermined, an inverse will not exist because there is not a matrix such that AA^-1 = A^-1A = I.
    

13. 1. -2     (-8) - (-6) = -2
       

    2.  140  (0 + 216 - 12) - (0 - 8 + 72) = 140
       

    3.  940  
       

    4. -120  
       

    
    

14. 1. Consistent and underdetermined because row 3 is a combination of the other two rows (because row 3 after elimination becomes all zeros).
       

    2. Consistent and uniquely determined because there is a single solution to the system. The matrix A is invertible.
       

    3. Inconsistent because after elimination, the last row is an unsolvable equation.
       

    4. Consistent and uniquely determined because there is a single solution to the system. The matrix A is invertible.