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Let the following be n*n square matrices

  1. Feb 3, 2005 #1
    Let the following be n*n square matrices: A, B, C, D
    | A B |
    | C D | = | A*D - C*B |

    If A*C = C*A (**).

    The proof:
    1) A is invertible.
    A' = Inverse(A).
    ( I 0 ) ( A B ) ( A B )
    ( -C*A' I ) * ( C D ) = ( 0 -C*A'*B+D )
    | A B | | A B |
    | C D | = | 0 -C*A'*B+D | = |A| * | -C*A'*B+D |
    = | A*D - A*C*A'B | = | A*D - C*B |
    2) Otherwise
    Replace A by A + epsilon*I.
    Now (**) is valid for the perturbated A too.
    We are in case (1) again. Take the limit epsilon -> 0.

    Problem:
    Can you avoid this analytic method?
    Do you think is there a pure algebraic derivation?
     
  2. jcsd
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