Home
Class 12
MATHS
Let A be a square matrix. Then prove tha...

Let `A` be a square matrix. Then prove that `(i) A + A^T` is a symmetric matrix,`(ii) A -A^T` is a skew-symmetric matrix and`(iii) AA^T` and `A^TA` are symmetric matrices.

Text Solution

Verified by Experts

i) Let `B=A+A^{T}`.
NOW,
`B^{T}`
`=(A+A^{T})^{T} `
`=A^{T}+(A^{T})^{T} `
`=A^{T}+A` [Since `(A^{T})^{T}=A`]
=B .
...
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 be a square matrix.Then prove that (i)A+A^(T) is a symmetric matrix,(ii) A-A^(T) is a skew-symmetric matrix and (iii)AA^(T) and A^(T)A are symmetric matrices.

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

A is symmetric matrix if A^T =

Let A be a square matrix.Then which of the following is not a symmetric matrix -

The inverse of a skew symmetric matrix is

A square matrix A is said skew - symmetric matrix if

If A is any square matrix,then A+A^(T) is skew symmetric

If A and B are symmetric matrices,prove that ABBA is a skew symmetric matrix.

If A is skew-symmetric matrix then A^(2) is a symmetric matrix.

Prove that inverse of a skew-symmetric matrix (if it exists) is skew-symmetric.

RD SHARMA-ALGEBRA OF MATRICES-Solved Examples And Exercises
  1. Show that the elements on the main diagonal of a skew-symmetric mat...

    Text Solution

    |

  2. If the matrix A=[0a3 2b-1c1 0] is skew-symmetric, find the values o...

    Text Solution

    |

  3. Let A be a square matrix. Then prove that (i) A + A^T is a symmetric m...

    Text Solution

    |

  4. Prove that every square matrix can be uniquely expressed as the sum of...

    Text Solution

    |

  5. If A and B are symmetric matrices, then show that A B is symmetric ...

    Text Solution

    |

  6. Show that the matrix B^T\ A B is symmetric or skew-symmetric accord...

    Text Solution

    |

  7. Let A and B be symmetric matrices of same order. Then A+B is a symmetr...

    Text Solution

    |

  8. Express the matrix A=[[3,2, 3],[ 4, 5, 3],[ 2, 4, 5]] as the sum of a ...

    Text Solution

    |

  9. Show that all positive integral powers of a symmetric matrix are sy...

    Text Solution

    |

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

    Text Solution

    |

  11. A matrix which is both symmetric as well as skew-symmetric is a nul...

    Text Solution

    |

  12. If A=[2 3 4 5] , prove that A-A^T is a skew-symmetric matrix.

    Text Solution

    |

  13. If A=[3-4 1-1] , show that A-A^T is a skew symmetric matrix.

    Text Solution

    |

  14. If the matrix A=[5 2x y z-3 4t-7] is a symmetric matrix, find x ,\ y ,...

    Text Solution

    |

  15. Let A=[3 2 7 1 4 3-2 5 8] . Find matrices X and Y such that X+Y=A , wh...

    Text Solution

    |

  16. Express the matrix A=[4 2-1 3 5 7 1-2 1] as the sum of a symmetric and...

    Text Solution

    |

  17. Define a symmetric matrix. Prove that for A=[2 4 5 6] , A+A^T is a ...

    Text Solution

    |

  18. Express the matrix A=[3-4 1-1] as the sum of a symmetric and a skew-sy...

    Text Solution

    |

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

    Text Solution

    |

  20. If A is an mxxn matrix and B is nxxp matrix does A B exist? If yes, wr...

    Text Solution

    |