Fsc Part 1 Mathematics (Complete Solution)

Q6
Show That

If A=[aij]3x3 , show that

(i) λ(μA)=(λμ)A

SOLUTION: λ(μA)=(λμ)A

L.H.S=λ(μA)

       =λ(μ[aij]3x3)

      = λ([μaij]3x3)

     =[λμaij]3x3

    =(λμ)[μaij]3x3

    =(λμ)A

    =R.H.S

Hence    λ(μA)=(λμ)A

(ii) (λ+μ)A=λA+μA

SOLUTION:   (λ+μ)A=λA+μA

L.H.S=(λ+μ)[aij]3x3

    =[(λ+μ)aij]3x3 

     =[λaij+μaij]3x3

     =[λaij]3x3+[μaij]3x3

    =λ[aij]3x3+μ[aij]3x3

   = λA+μA

  =R.H.S

Hence   (λ+μ)A=λA+μA

(iii) λA-A=(λ-1)A

SOLUTION:  λA-A=(λ-1)A

L.H.S=λA-A

     = λ[aij]3x3 - 1[aij]3x3

     = (λ-1)[aij]3x3 

     = (λ-1)A

     =R.H.S

Hence  λA-A=(λ-1)A

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