Page 326 - 35Linear Algebra
P. 326

326                                                                       Sample First Midterm


                                   (e) This amounts to explicitly performing the matrix manipulations


                                                            MX − V, MY 1 , and MY 2
                                       to verify that they are all zero vectors.

                               3. As usual, be sure to write out the row operations above the ∼’s so your work
                                  can be easily checked.

                                                                                    
                                                          1   2   3   4 1 0 0 0
                                                         2   4   7 11 0 1 0 0       
                                                                                    
                                                          3   7 14 25 0 0 1 0
                                                                                    
                                                          4 11 25 50 0 0 0 1
                                                                                    
                                                         1 2     3   4    1 0 0 0
                                                        0 0     1   3 −2 1 0 0      
                                                    ∼                               
                                                         0 1     5 13 −3 0 1 0
                                                                                    
                                                         0 3 13 34 −4 0 0 1
                                                                                      
                                                       1 0 −7 −22         7 0 −2 0
                                                       0 1     5    13 −3 0       1 0
                                                                                      
                                                  ∼                                   
                                                      0 0     1     3 −2 1       0 0  
                                                       0 0 −2      −5     5 0 −3 1
                                                                                     
                                                        1 0 0 −1 −7          7 −2 0
                                                       0 1 0 −2        7 −5     1 0  
                                                   ∼                                 
                                                        0 0 1      3 −2      1   0 0
                                                                                     
                                                        0 0 0      1    1    2 −3 1
                                                                                     
                                                       1 0 0 0 −6         9 −5      1
                                                       0 1 0 0       9 −1 −5        2
                                                                                     
                                                  ∼                                    .
                                                      0 0 1 0 −5 −5           9 −3   
                                                       0 0 0 1       1    2 −3      1
                                  Check

                                                                                          
                                           1   2   3   4     −6    9 −5      1       1 0 0 0
                                           2   4   7 11       9 −1 −5        2       0 1 0 0
                                                                                          
                                                                             =              .
                                          3   7 14 25     −5 −5      9 −3       0 0 1 0   
                                           4 11 25 50         1    2 −3      1       0 0 0 1

                               4.
                                                                           !
                                                                   1    1       11   4
                                                           2    3                      −
                                                 T
                                                     −1
                                               M M     =             5    5   =    5 2  5   .
                                                                                        3
                                                           1 −1      3  − 2       −
                                                                     5    5        5    5
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