Home
Class 12
MATHS
If a and B are non-singular symmetric ma...

If a and B are non-singular symmetric matrices such that `AB=BA`, then prove that `A^(-1) B^(-1)` is symmetric matrix.

Text Solution

Verified by Experts

We have `AB=BA`
Now, `(A^(-1) B^(-1))^(T)=((BA)^(-1))^(T)=((BA)^(T))^(-1)`
`=(A^(T)B^(T))^(-1)=(AB)^(-1)=(BA)^(-1)=A^(-1)B^(-1)`
Thus, `A^(-1) B^(-1)` is symmetric matrix.
Promotional Banner

Topper's Solved these Questions

  • MATRICES

    CENGAGE|Exercise Exercise 13.1|5 Videos

  • MATRICES

    CENGAGE|Exercise Exercise 13.2|6 Videos

  • MATRICES

    CENGAGE|Exercise Multiple Correct Answer|7 Videos

  • MATHMETICAL REASONING

    CENGAGE|Exercise JEE Previous Year|10 Videos

  • METHODS OF DIFFERENTIATION

    CENGAGE|Exercise Multiple Correct Answer Type|7 Videos

Similar Questions

Explore conceptually related problems

If A and B are square matrices such that AB = BA then prove that A^(3)-B^(3)=(A-B) (A^(2)+AB+B^(2)) .

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

If A and B are symmetric matrices of same order, prove that AB + BA is a symmetric matrix.

If A and B are symmetric matrices of same order, prove that AB-BA is a skew -symmetric matrix.

If A and B are symmetric matrices of same order, prove that AB+BA is a symmetric matrix.

If A and B are two square matrices such that B=-A^(-1)BA , then (A+B)^(2) is equal to

Let A and B be two symmetric matrices. Prove that AB= BA if and only if AB is a symmetric matrix.

If A and B are square matrices of the same order such that AB = BA, then prove by induction that AB^(n)=B^(n)A . Further, prove that (AB)^(n)=A^(n)B^(n) for all n in N .

If A and B are symmetric matrices, prove that AB-BA is a skew symmetric matrix.

Let a be square matrix. Then prove that A A^(T) and A^(T) A are symmetric matrices.

CENGAGE-MATRICES-Examples
  1. Matrices a and B satisfy AB=B^(-1), where B=[(2,-1),(2,0)]. Find (i...

    Text Solution

    |

  2. Given the matrices a and B as A=[(1,-1),(4,-1)] and B=[(1,-1),(2,-2)]....

    Text Solution

    |

  3. If M is the matrix [(1,-3),(-1,1)] then find matrix sum(r=0)^(oo) ((-1...

    Text Solution

    |

  4. Let p be a non singular matrix, and I + P + p^2 + ... + p^n = 0, then ...

    Text Solution

    |

  5. If A and B are square matrices of same order such that AB=O and B ne O...

    Text Solution

    |

  6. If A is a symmetric matrix, B is a skew-symmetric matrix, A+B is nonsi...

    Text Solution

    |

  7. If the matrices, A, B and (A+B) are non-singular, then prove that [A(A...

    Text Solution

    |

  8. If matrix a satisfies the equation A^(2)=A^(-1), then prove that A^(2^...

    Text Solution

    |

  9. If a and B are non-singular symmetric matrices such that AB=BA, then p...

    Text Solution

    |

  10. If A is a matrix of order n such that A^(T)A=I and X is any matric suc...

    Text Solution

    |

  11. Show that two matrices A=[(1,-1,0),(2,1,1)] and B=[(3,0,1),(0,3,1)] ...

    Text Solution

    |

  12. Using elementary transformations, find the inverse of the matrix : ...

    Text Solution

    |

  13. Let a be a 3xx3 matric such that [(1,2,3),(0,2,3),(0,1,1)]=[(0,0,1),...

    Text Solution

    |

  14. Using matrix method, solve the following system of equations: x+2y+z...

    Text Solution

    |

  15. Using matrix method, show that following system of equation is inconsi...

    Text Solution

    |

  16. about to only mathematics

    Text Solution

    |

  17. Find the characteristic roots of the two-rowed orthogonal matrix [(cos...

    Text Solution

    |

  18. Show that if lambda(1), lambda(2), ...., lamnda(n) are n eigenvalues o...

    Text Solution

    |

  19. If A is nonsingular, prove that the eigenvalues of A^(-1) are the reci...

    Text Solution

    |

  20. If one of the eigenvalues of a square matrix a order 3xx3 is zero, the...

    Text Solution

    |