Home
Class 12
MATHS
If A and P are the square matrices of th...

If A and P are the square matrices of the same order and if P be invertible, show that the matrices A and `P^(-1)` have the same characteristic roots.

Text Solution

Verified by Experts

Let `P^(-1)AP=B`
`therefore " " |B-lambdaI|=|P^(-1)AP-lambdaI\|`
`=|P^(-1)AP-P^(-1)lambdaP|`
`=|P^(-1)(A-lambdaI)p|`
`=P^(-1)||A-lambdaI||P|`
`=(1)/(|P|)|A-lambda||P|=|A-lambdaI|`
Promotional Banner

Topper's Solved these Questions

  • MATRICES

    ARIHANT MATHS|Exercise Exercise For Session 1|9 Videos

  • MATRICES

    ARIHANT MATHS|Exercise Exercise For Session 2|19 Videos

  • MATHEMATICAL INDUCTION

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|2 Videos

  • MONOTONICITY MAXIMA AND MINIMA

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|31 Videos

Similar Questions

Explore conceptually related problems

If A and B are two square matrices of the same order, then AB=BA .

If A and B are symmetric matrices of the same order, then

If A and B are two square matrices of the same order, then A+B=B+A .

If A and B are two matrices of the same order, then A-B=B-A .

If A and B are square matrices of the same order, then (A+B) (A-B) is equal to

If A and B are square matrices of the same order, compute (A+B). (A-B)

If A and B are square matrices of the same order and A is symmetric, then show that B^t AB is also symmetric.

If A and B are symmetric matrices of same order then AB + BA is a :

If A and B are symmetric matrices of same order then AB - BA is a :

If A and B are invertible matrices of the same order, then (AB)^-1 is equal to

ARIHANT MATHS-MATRICES -Exercise (Questions Asked In Previous 13 Years Exam)
  1. If A and P are the square matrices of the same order and if P be inver...

    Text Solution

    |

  2. Let A=[(1,0,0),(0,1,1),(0,-2,4)],I=[(1,0,0),(0,1,0),(0,0,1)] and A^-1=...

    Text Solution

    |

  3. Evluate int 3x^2 dx

    Text Solution

    |

  4. If A=[(1,0),(1,1)] and I=[(1,0),(0,1)] then which one of the following...

    Text Solution

    |

  5. If A^(2)-A+I=O, then A^(-1) is equal to

    Text Solution

    |

  6. Let {:A=[(1,0,0),(2,1,0),(3,2,1)]:}and U1,U2,U3 be column matrices sat...

    Text Solution

    |

  7. Let A = [(1,0,0), (2,1,0), (3,2,1)], and U1, U2 and U3 are columns of ...

    Text Solution

    |

  8. If A= ((1,0,0),(2,1,0),(3,2,1)), U(1), U(2), and U(3) are column matri...

    Text Solution

    |

  9. Let A=[{:(1,2),(3,4):}]and B = [{:(a,0),(0,b):}] where a, b are natura...

    Text Solution

    |

  10. If A and B are square matrices of size nxxn such that A^2-B^2 = (A-B)(...

    Text Solution

    |

  11. Let A= [[5,5alpha,alpha],[0,alpha,5alpha],[0,0,5]] . If |A^2|...

    Text Solution

    |

  12. Let A and B be 3xx3 matrtices of real numbers, where A is symmetric, "...

    Text Solution

    |

  13. Let A be a square matrix all of whose entries are integers. Then wh...

    Text Solution

    |

  14. Let A be a 2xx2 matrix with real entries. Let I be the 2xx2 identi...

    Text Solution

    |

  15. Let A be the set of all 3xx3 symmetric matrices all of whose either 0 ...

    Text Solution

    |

  16. Let A be the set of all 3xx3 symmetric matrices all of whose either 0 ...

    Text Solution

    |

  17. The number of 3xx3 matrices A whose are ether 0 or 1 and for which t...

    Text Solution

    |

  18. Let A be a 2xx2 matrix Statement -1 adj (adjA)=A Statement-2 abs(a...

    Text Solution

    |

  19. The number of 3xx3 matrices a whose entries are either 0 or 1 and for ...

    Text Solution

    |

  20. Let P be an odd prime number and T(p) be the following set of 2xx2 mat...

    Text Solution

    |

  21. Let p be an odd prime number and T(P) be the following set of 2xx2 m...

    Text Solution

    |