Page 332 - 35Linear Algebra
P. 332

332                                                                    Sample Second Midterm


                                                  3
                                  the vector X ∈ R . Find, but do not solve, the characteristic polynomial of
                                  A.

                               3. Let M be any 2 × 2 matrix. Show

                                                                  1        1
                                                                       2
                                                                                  2
                                                       det M = − trM + (trM) .
                                                                  2        2
                                                              i
                               4. The permanent: Let M = (M ) be an n×n matrix. An operation producing
                                                              j
                                  a single number from M similar to the determinant is the “permanent”
                                                               X     1    2        n
                                                    perm M =      M σ(1) M σ(2)  · · · M σ(n)  .
                                                                σ

                                  For example

                                                                 a  b
                                                          perm         = ad + bc .
                                                                 c d
                                  Calculate
                                                                          
                                                                    1 2 3
                                                             perm   4 5 6   .
                                                                    7 8 9

                                  What do you think would happen to the permanent of an n × n matrix M
                                  if (include a brief explanation with each answer):


                                   (a) You multiplied M by a number λ.
                                   (b) You multiplied a row of M by a number λ.

                                   (c) You took the transpose of M.
                                   (d) You swapped two rows of M.

                               5. Let X be an n × 1 matrix subject to


                                                                  T
                                                                X X = (1) ,
                                  and define
                                                                           T
                                                              H = I − 2XX ,
                                  (where I is the n × n identity matrix). Show

                                                                    T
                                                              H = H = H    −1 .

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