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Let A = ( 0 1 0 1 0 0 0 0 1 ) .  Then the number of 3 × 3 matrices B with entries from the set {1,2,3,4,5} and satisfying AB = BA is…………..

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    Answered by

    alok kumar singh | Contributor-Level 10

    a month ago

    A = ( 0 1 0 1 0 0 0 0 1 ) 3 × 3

     B is a matrix of same order with entries from {1,2,3,4,5}. and satisfying AB = BA.

    L e t     B = ( a 1 1 a 1 2 a 1 3 a 2 1 a 2 2 a 2 3 a 3 1 a 3 2 a 3 3 )           

    ∴ ( 0 1 0 1 0 0 0 0 1 ) ( a 1 1 a 1 2 a 1 3 a 2 1 a 2 2 a 2 3 a 3 1 a 3 2 a 3 3 ) = ( a 1 1 a 1 2 a 1 3 a 2 1 a 2 2 a 2 3 a 3 1 a 3 2 a 3 3 ) ( 0 1 0 1 0 0 0 0 1 )

    a 2 1 = a 1 2 , a 2 2 = a 1 1 , a 2 3 = a 1 3 , a 3 1 = a 3 2

    ∴ there exist only 5 distinct entries in the matrix B so that possible case = 55 = 3125

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