32 x 1 2 x 2 x 4 4 3 x 2 x 3 2 x 4 3 4 x 1 x 3 2 x 4 2 x 1 x 3 3 x 4 1 2 For

# 32 x 1 2 x 2 x 4 4 3 x 2 x 3 2 x 4 3 4 x 1 x 3 2 x 4

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32. x 1 +2 x 2 + x 4 = 4 , 3 x 2 + x 3 +2 x 4 = 3 , 4 x 1 - x 3 +2 x 4 = 2 , x 1 + x 3 +3 x 4 = 1 . 2 For each of the following systems, find h so that the system has solutions. 33. x 1 +2 x 2 +3 x 3 = 5 , 2 x 1 - x 2 - 2 x 3 = 2 h, 3 x 1 - x 2 - x 3 = h. 34. x 1 + x 2 + x 3 = 2 , 2 x 1 +3 x 2 +2 x 3 = 5 , x 1 + x 2 +( h 2 - 5) x 3 = h. 35. x 3 +2 x 4 = 1 , - 2 x 1 +2 x 2 - x 3 +4 x 4 = h, x 1 - x 2 + x 3 - x 4 = 1 . 36. x 1 + x 2 + x 3 + x 4 = 1 , x 1 + hx 4 = 1 , x 2 + hx 3 = 1 . 37. For what value(s) of h does the following system has solution for any b 1 , b 2 , b 3 . For such value(s) of h , solve the system by the method of reduction. x 1 + hx 3 + x 4 = b 1 , - x 1 + h 2 x 3 - (2 + h ) x 4 = b 2 , x 1 + x 2 +2 x 4 = b 3 . 38. Consider A = 0 0 1 2 - 2 2 - 1 4 1 - 1 1 - 1 , b = 1 h 1 . For what h , does Ax = b has solution? Find the general solution of the homogeneous system Ax = O . 39.Does the following homogeneous system has a nontrivial solution? 40. Without performing any computation, justify that the system below has a nontrivial solution. x 1 - 3 x 2 + 9 x 3 - 2 x 4 + 3 x 5 = 0 , 3 x 1 + 6 x 2 + 4 x 3 - 3 x 4 - 2 x 5 = 0 , - x 1 + 2 x 2 - 7 x 3 + 5 x 4 + 7 x 5 = 0 , x 1 - 5 x 2 + 4 x 3 - 3 x 4 - 8 x 5 = 0 . 132
6.4 Rank 6.4 Rank 4 Definition (Rank) The rank of a matrix A , written rank A , is equal to the number of pivots in a row echelon form of A . ¥ Example 6.4.1 (Rank) In Examples 6.2.1, 6.2.2, 6.2.3 and 6.2.4 (page 121) , we saw that the following augmented matrices 2 - 1 - 1 - 1 1 2 , 1 1 1 1 2 3 2 1 3 8 2 2 , 1 1 1 2 1 3 - 1 4 0 1 - 1 2 , 1 - 1 3 3 3 1 1 5 0 1 - 2 - 1 , which have the following row echelon forms " / £ ¡ ¢ - 1 1 2 0 / £ ¡ ¢ 1 3 # , / £ ¡ ¢ 1 1 1 1 0 / £ ¡ ¢ 1 0 - 1 0 0 / £ ¡ ¢ - 1 4 , / £ ¡ ¢ 1 1 1 2 0 / £ ¡ ¢ 1 - 1 1 0 0 0 / £ ¡ ¢ 1 , / £ ¡ ¢ 1 - 1 3 3 0 / £ ¡ ¢ 1 - 2 - 1 0 0 0 0 . The ranks of the four augmented matrices are 2, 3, 3, 2, respectively. 2 A system Ax = b of linear equations may contain many equations. However, after Gaussian elimination, some equations become the trivial equation 0 = 0. Therefore the rank of £ A b / is the “essential number” of equations, or the “true size” of the system. ¥ Example 6.4.2 (Rank) There seems to be many equations in the following system x 1 - x 2 = 1 , 2 x 1 - 2 x 2 = 2 , - x 1 + x 2 = - 1 , 5 x 1 - 5 x 2 = 5 . However, all the equations can be deduced from the first one. There is essentially only one equation (i.e., x 1 - x 2 = 1) in the system. Correspondingly, the rank of the matrix 1 - 1 1 2 - 2 2 - 1 1 - 1 5 - 5 5 is 1. In fact, one may verify this by finding its row echelon form 1 - 1 1 2 - 2 2 - 1 1 - 1 5 - 5 5 - 2 R 1 + R 2 , R 1 + R 3 ------------→ - 5 R 1 + R 4 / £ ¡ ¢ 1 - 1 1 0 0 0 0 0 0 0 0 0 . 2 Rank may be used to describe the existence and uniqueness for solutions of a system Ax = b . Note that if we delete the last column of the row echelon form of £ A b / , then we get the row echelon form of A . Therefore by the discussion of Section 6.3.3 (page 128) , we have rank £ A b / = ( rank A , if Ax = b has solutions , rank A + 1 , if Ax = b has no solution .