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

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

Text Solution

Verified by Experts

We have, `(A A^(T))^(T)=(A^(T))^(T)A^(T)=A A^(T)`
Thus, `A A^(T)` is symmetric.
Similarly, it can be proved that `A^(T)A` is symmetric.
Promotional Banner

Topper's Solved these Questions

  • MATHMETICAL REASONING

    CENGAGE PUBLICATION|Exercise Archives|10 Videos

  • METHODS OF DIFFERENTIATION

    CENGAGE PUBLICATION|Exercise Multiple Correct Answer Type|7 Videos

Similar Questions

Explore conceptually related problems

Let A be a square matrix. Then prove that (i) A + A^T is a symmetric matrix and, (ii) A -A^T is a skew-symmetric matrix

If A and B are symmetric matrices then

Let A be any 3xx2 matrix. Then prove that det. (A A^(T))=0 .

Let A and B two symmetric matrices of order 3. Statement 1 : A(BA) and (AB)A are symmetric matrices. Statement 2 : AB is symmetric matrix if matrix multiplication of A with B is commutative.

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

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

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

Let A and B be two square matrices of the same size such that AB^(T)+BA^(T)=O . If A is a skew-symmetric matrix then BA is

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

If A and B are matrices of the same order, then A B^T-B A^T is a (a) skew-symmetric matrix (b) null matrix (c) unit matrix (d) symmetric matrix

CENGAGE PUBLICATION-MATRICES-All Questions
  1. Let A=[(2,1),(0,3)] be a matrix. If A^(10)=[(a,b),(c,d)] then prove th...

    Text Solution

    |

  2. Show that the solutions of the equation [(x,y),(z,t)]^2=0 are[(x,y),(...

    Text Solution

    |

  3. Let A be square matrix. Then prove that A A^(T) and A^(T) A are symmet...

    Text Solution

    |

  4. If A, B are square materices of same order and B is a skewsymmetric ma...

    Text Solution

    |

  5. If A and B are square matrices of same order such that AB+BA=O, then p...

    Text Solution

    |

  6. Let A=[(1,2),(-1,3)] .If A^6=kA-205I then find the numerical quantit...

    Text Solution

    |

  7. Let A, B, C, D be (not necessarily square) real matrices such that A^T...

    Text Solution

    |

  8. If A and B are square matrices of the same order such that A B = B A,...

    Text Solution

    |

  9. If A=[-1 1 0-2] , then prove that A^2+3A+2I=Odot Hence, find Ba n dC m...

    Text Solution

    |

  10. If A=[(3,-4),(1,-1)] then find tr. (A^(2012)).

    Text Solution

    |

  11. If A is a nonsingular matrix satisfying AB-BA=A, then prove that det. ...

    Text Solution

    |

  12. If det, (A-B) ne 0, A^(4)=B^(4), C^(3) A=C^(3)B and B^(3)A=A^(3)B, the...

    Text Solution

    |

  13. Given a matrix A=[(a,b,c), (b,c,a), (c,a,b)],where a ,b ,c are real po...

    Text Solution

    |

  14. If M is a 3xx3 matrix, where det M=1a n dM M^T=1,w h e r eI is an iden...

    Text Solution

    |

  15. Consider point P(x, y) in first quadrant. Its reflection about x-axis ...

    Text Solution

    |

  16. If A=[[2,-2,-4],[-1,3,4],[1,-2,-3]] then A is 1) an idempotent matrix ...

    Text Solution

    |

  17. If A= [(1,1,3),(5,2,6),(-2,-1,-3)] then find A^(14)+3A-2I

    Text Solution

    |

  18. The matrix A=[-5-8 0 3 5 0 1 2-] is a. idempotent matrix b. involut...

    Text Solution

    |

  19. If abc=p and A=[(a,b,c),(c,a,b),(b,c,a)], prove that A is orthogonal i...

    Text Solution

    |

  20. Let A be an orthogonal matrix, and B is a matrix such that AB=BA, then...

    Text Solution

    |