Let's suppose A , B are n −order matrix, satisfying A B = B A = O, r ( A ) = r ( A 2 ), prove that r ( A + B ) = r ( A ) + r ( B )

Question

Let&#039;s suppose   A  ,  B are   n  −order matrix, satisfying   A  B  =  B  A  =  O,   r  (  A  )  =  r  (      A    2    ), prove that  r  (  A  +  B  )  =  r  (  A  )  +  r  (  B  )

Zayden Andrade · Accepted Answer

Let   f be a linear operator of a linear space   L with matrix   A in some basis of   L.If   M  =  f  (  L  ), then   f  (  M  )  =  M. (Use here the equality   r  (  A  )  =  r  (      A    2    ).)It follows that the matrix   A  is conjugate to a block matrix of the form      (                                        A                      11                                                A                      12                                                0                          0                      )    ,where   |      A      11    |  ≠  0. (What are the dimensions of the blocks?)Let  B  =      (                                        B                      11                                                B                      12                                                            B                      21                                                B                      22                                )    .Since   A  B  =  B  A  =  0, then       B      11    =  0,       B      12    =  0,       B      21    =  0.Now you can easily prove your equality.

Gaaljh · Accepted Answer

Thanks, that&#039;s a great method. But I hope we may not introduce any concept of linear operaotr at all. I am searching for a method that only uses elementary block matrix.

Let's suppose A , B are n − order matrix, satisfying A B

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