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                                   T
                          (h) Is M M diagonalizable?
                                      T
                          (i) Does M M have a zero eigenvalue?
                          (j) Suppose U = V and ker L 6= {0 U }. Find an eigenvalue of M.
                          (k) Suppose U = V and ker L 6= {0 U }. Find det M.

                      4. Consider the system of equations

                                            x +     y   +   z   +   w   = 1
                                            x + 2y + 2z + 2w = 1
                                            x + 2y + 3z + 3w = 1

                         Express this system as a matrix equation MX = V and then find the solution
                         set by computing an LU decomposition for the matrix M (be sure to use
                         back and forward substitution).

                      5. Compute the following determinants

                                                                                  
                                                                      1   2   3   4
                                                           
                                                    1 2 3

                                       1 2                           5   6   7   8 
                                  det         , det   4 5 6  , det                ,
                                       3 4                           9 10 11 12   
                                                    7 8 9
                                                                     13 14 15 16
                                                                      
                                                     1    2   3   4   5
                                                     6    7   8   9 10
                                                                      
                                                                      
                                               det   11 12 13 14 15    .
                                                                      
                                                    16 17 18 19 20
                                                                      
                                                    21 22 23 24 25
                         Now test your skills on
                                                                                   
                                             1           2           3      · · ·  n
                                          n + 1       n + 2       n + 3    · · · 2n  
                                                                                   
                                                                                    
                                         2n + 1      2n + 2      2n + 3         3n   .
                                  det 
                                             .
                                             .                               .
                                                                           .     . 
                                            .                                .   . . 
                                                      2
                                                                  2
                                          2
                                        n − n + 1 n − n + 2 n − n + 3 · · · n     2
                         Make sure to jot down a few brief notes explaining any clever tricks you use.
                      6. For which values of a does
                                                                 
                                                    
                                                   1         1     a 
                                                                              3
                                                                    1
                                                         ,
                                                      0
                                         U = span          2        = R ?
                                                                 ,
                                                      1     −3      0
                                                                      
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