Question 5 Please explain your answer Thanks In this questio

Question 5

Please explain your answer,

Thanks!

In this question, A is a square matrix of the order nxn. In sections II and III the matrix A\' is the result of swapping the rows i and j in matrix A. i is different from j. Prove or disprove the following proposition: I. Assume that A notequalto I_n, A notequalto 0, A^2 = A is true for A. Then the equation Ax = 0 has only the trivial solution. II. The system (A + A\')x_= 0 has infinitely many solutions. III. If the system (AA\')x = 0 has infinitely many solutions, then the system Ax = 0 has infinitely many solutions.

Solution

1 . A²  = A , A X A - A = O applying distributive property A ( A - I ) = O or ( A - I ) A = O . operating X on either side ( A - I ) A x =O

   ^{If X is not = O then either A = O} or A - I = O both are not true . hence X is a null column matrix

2 A \' is a matrix obtained by interchanging i th and j th rows. hence In tne matrix A + A\' i th and j th row are equal .

   In A + A\' i th row a_i1 + a _{j1 ,} a_{i2 +} a_{j2 ------- which is same as the j th row .}

   in a matri x if 2 rows are identicl the the system has infinite number of solution

   when we solve ( A + A\') X =O the number of equations is lesser than the number of unknowns in the matrix X . that type of system has infinite number of solutions