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Suppose A and B be two ono-singular matr...

Suppose A and B be two ono-singular matrices such that
`AB= BA^(m), B^(n) = I and A^(p) = I `, where `I` is an identity matrix.
If `m = 2 and n = 5 ` then p equals to

A

30

B

31

C

33

D

81

Text Solution

Verified by Experts

The correct Answer is:
B
`because AB = BA^(m) `
` rArr B = A^(-1) BA^(m)`
`therefore B^(n) = underset("n times")(underbrace((A^(-1) BA^(m))(A^(-1)BA^(m))... (A^(-1) BA^(m))))`
`=A^(-1) underset("n times")(underbrace(BA^(m-1)BA^(m-1)... BA^(m-1)BA^(m-1)))A` ...(i)
Given, ` AB = BA^(m)`
` rArr A AB = ABA^(m) = BA^(2m) rArr A A AB = BA^(3m)`
Similarly, `A^(x) B = BA^(mx) AA m in N`
From Eq. (i) we get
`B^(n)=A^(-1) BA^(m-1) underset("(n-1) times")(underbrace(BA^(m-1)BA^(m-1)... BA^(m-1)BA^(m-1)))A`
`=A^(-1) B(A^(m-1) B)A^(m-1) underset("(n-2) times")(underbrace(BA^(m-1)... BA^(m-1)BA^(m-1)))A`
`=A^(-1) BBA^((m-1)m) A^(m-1) underset("(n-2) times")(underbrace(BA^(m-1)... BA^(m-1)BA^(m-1)))A`
`=A^(-1) B^(2)A^((m^(2)-1)) underset("(n-2) times")(underbrace(BA^(m-1)... BA^(m-1)BA^(m-1)))A`
`..." " ..." "... " "... `
`=A^(-1) B^(m) (A) ^(m^(n)-1)A`
`I = A^(-1) I A^(m^(n)-1) A [because B^(n) = I]`
`I = A^(-1) A^(m^(n)-1) A= A^(-1) A^(m^(n))`
`rArr I = A^(m^(n)-1)`
`therefore p= m^(n)-1 " "...(ii) [because A^(p) = I]`
Put `m = 2, n = 5 ` in Eq. (ii), we get
`p= 2^(5) - 1 = 31`
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Knowledge Check

  • Suppose A and B be two ono-singular matrices such that AB= BA^(m), B^(n) = I and A^(p) = I , where I is an identity matrix. The relation between m, n and p, is

    A
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    B
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    C
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    D
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  • Suppose A and B be two ono-singular matrices such that AB= BA^(m), B^(n) = I and A^(p) = I , where I is an identity matrix. Which of the following orderd triplet (m, n, p) is false?

    A
    `(3, 2, 80)`
    B
    `(6, 3, 215)`
    C
    `(8, 3, 510)`
    D
    `(2, 8, 255)`

  • Let A and B are two non - singular matrices such that AB=BA^(2),B^(4)=I and A^(k)=I , then k can be equal to

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