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Homework Help: Matrix properties

  1. Jan 30, 2010 #1
    1. The problem statement, all variables and given/known data

    If B=C^(-1)

    Is (B+C)^2=B^2+2BC+C^2

    2. Relevant equations

    If A and B are (mXn) matrices and C is an (nXp) matrix, then (A+B)C=AC+BC

    If A is an mXn matrix and B and c are nXp matrices,then A(B+C)=AB+AC


    3. The attempt at a solution


    (B+C)(B+C)=B^2+2BC+C^2

    Then I decided to substitute B+C for D, but only one of them.

    D(B+C)=B^2+2BC+C^2

    DB+DC=B^2+2BC+C^2

    Then I substituted the b+c back in for D

    (B+C)B+(B+C)C=B^2+2BC+C^2

    Then I get

    BB+CB+BC+CC=B^2+2BC+C^2

    Then from here I plugged in the C^-1 in for the B's in the two middle terms, which gives c time c inverse, plust c inverse times C, and they both are equal to 1, and 1 plus 1 is equal to two so I get

    B^2+2+C^2=B^2+2BC+C^2

    And then on the right hand side you can plug the c inverse in for b and you get 2 times c inverse times C which is just equal to 2.


    B^2+2+C^2=B^2+2+C^2

    which finall gets me back to

    (B+C)^2=B^2+2BC+C^2

    B^2+2+C^2=B^2+2+C^2

    Did I screw up anywhere? The thing I am mainly not sure of is if I can make the substution like that.
     
  2. jcsd
  3. Jan 31, 2010 #2

    HallsofIvy

    User Avatar
    Science Advisor

    You can make any substitution you like! And your proof is correct though more complicated than needed.

    For any B and C, [itex](B+ C)^2= B^2+ BC+ CB+ C^2[/itex]. Since we are given that [itex]B= C^{-1}[/itex], both BC and CB are equal to I. That is [itex](B+ C)^2= B^2+ 2BC+ C^2[/itex] because both are equal to [itex]B^2+ 2I+ C^2[/itex].
     
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