Home
Class 12
MATHS
Let A ,B be two matrices such that they ...

Let `A ,B` be two matrices such that they commute. Show that for any positive integer `n ,` `A B^n=B^n A` (ii) `(A B)^n=A^nB^n`

Text Solution

Verified by Experts

Here, `a `and `B` are the matrices such that they commute.
`:. AB = BA`.
(i) Now, `AB^n = A*B*B*B...B`
`= B*A*B*B...*B`
`= B*B*A*B...*B`
If we go this way, it will become,
`AB^n = B*B*B...*B*A`
`=>AB^n = B^nA`

...
Promotional Banner

Topper's Solved these Questions

  • ADJOINTS AND INVERSE OF MATRIX

    RD SHARMA|Exercise QUESTION|1 Videos

  • ALGEBRA OF VECTORS

    RD SHARMA|Exercise Solved Examples And Exercises|331 Videos

Similar Questions

Explore conceptually related problems

Let A,B be two matrices such that they commute.Show that for any positive integer n,AB^(n)=B^(n)A

Let A,B be two matrices such that they commute.Show that for any positive integer n,(AB)^(n)=A^(n)B^(n)

Let A,B be two matrices such that they commute,then for any positive integer nAB^(n)=B^(n)A(AB)^(n)A^(n)B^(n) only (i) and (ii) correct both (i) and (ii) correct only (ii) is correct none of (i) and (ii) is correct

Let A and B are two matrices such that AB = BA, then for every n in N

If A and B are two matrices such that AB=BA, then for every n epsilonN (A) (AB)^n=A^nB^n (B) A^nB=BA^n (C) (A^(2n)-B^(2n))=(A^n-B^n)(A^n+B^n) (D) (A+B)^n=^nC_0A^n+^nC_1A^(n-1)B+^nC_nB^n

Let A and B any two n xx n matrices such that the following conditions hold : AB= BA and there exist positive integers k and l such that A^(k)=I (the identity matrix) and B^(l)=0 (the zero matrix). Then-

If n is an even positive integer, then a^(n)+b^(n) is divisible by

If n is an odd positive integer, then a^(n)+b^(n) is divisible by

RD SHARMA-ALGEBRA OF MATRICES-Solved Examples And Exercises
  1. Find the values of xa n dy if [x+10 y^2+2y0-4]=[3x+4 3 0y^2-5y]

    Text Solution

    |

  2. For what values of x and y are the following matrices equal ? A=[(2x...

    Text Solution

    |

  3. Let A ,B be two matrices such that they commute. Show that for any pos...

    Text Solution

    |

  4. Let A=[[2,3],[-1,2]] and f(x)=x^2 this result of find A^5

    Text Solution

    |

  5. If A=[3 5],B=[7 3], then find a non-zero matrix C such that AC=AC.

    Text Solution

    |

  6. If A is a square matrix such that A^2=A , show that (I+A)^3=7A+Idot

    Text Solution

    |

  7. Under what conditions is the matrix A^2-B^2=(A-B)(A+B) is true?

    Text Solution

    |

  8. If a is a non-zero real or complex number. Use the principle of mathe...

    Text Solution

    |

  9. If A = [[cos^2theta, costhetasintheta],[costhetasintheta, sin^2theta]]...

    Text Solution

    |

  10. If A and B are square matrices of order n , then prove that Aa n dB wi...

    Text Solution

    |

  11. Show that positive odd integral powers of a skew-symmetric matrix are ...

    Text Solution

    |

  12. If A=[(costheta,-sintheta),(sintheta,costheta)], then find the values...

    Text Solution

    |

  13. If A=d i ag(abc), show that A^n=d i ag(a^nb^nc^n) for all positive int...

    Text Solution

    |

  14. If A=[[costheta,isintheta],[isintheta,costheta]], then prove by princ...

    Text Solution

    |

  15. Let A=[[1,-1, 0],[ 2, 1, 3],[ 1, 2, 1]] a n d B=[[1 ,2, 3],[ 2, 1, 3],...

    Text Solution

    |

  16. If [xy ,4 , z+6 , x+y]=[8,w,0, 6], write the value of (x+y+z)dot

    Text Solution

    |

  17. If A=[a(i j)] is a skew-symmetric matrix, then write the value of sumi...

    Text Solution

    |

  18. Express the matrix [3-2-4 3-2-5-1 1 2] as the sum of a symmetric and s...

    Text Solution

    |

  19. Show that the elements on the main diagonal of a skew-symmetric matrix...

    Text Solution

    |

  20. If A is a skew-symmetric and n in N such that (A^n)^T=lambdaA^n , wri...

    Text Solution

    |