Page 331 - 35Linear Algebra
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                                                        Sample Second Midterm






                   Here are some worked problems typical for what you might expect on a second
                   midterm examination.

                                                                                        a   b
                      1. Determinants: The determinant det M of a 2 × 2 matrix M =             is
                                                                                        c d
                         defined by
                                                    det M = ad − bc .

                          (a) For which values of det M does M have an inverse?
                          (b) Write down all 2×2 bit matrices with determinant 1. (Remember bits
                              are either 0 or 1 and 1 + 1 = 0.)
                          (c) Write down all 2 × 2 bit matrices with determinant 0.
                          (d) Use one of the above examples to show why the following statement is
                              FALSE.
                                  Square matrices with the same determinant are always row
                                  equivalent.

                      2. Let
                                                                  
                                                           1 1 1
                                                                  
                                                   A =    2 2 3     .
                                                                  
                                                           4 5 6
                                                                                          
                                                                                         1
                         Compute det A. Find all solutions to (i) AX = 0 and (ii) AX =    2    for
                                                                                         3


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