Home
Class 12
MATHS
If the matrices, A, B and (A+B) are non-...

If the matrices, A, B and `(A+B)` are non-singular, then prove that `[A(A+B)^(-1) B]^(-1) =B^(-1)+A^(-1)`.

Text Solution

Verified by Experts

`[A(A+B)^(-1) B]^(-1) =B^(-1) ((A+B)^(-1))^(-1) A^(-1)`
`=B^(-1) (A+B)A^(-1)`
`=(B^(-1) A+B^(-1)B) A^(-1)`
`=(B^(-1)A+I)A^(-1)`
`=B^(-1) A A^(-1)+IA^(-1)`
`=B^(-1)I+A^(-1)`
`=B^(-1)+A^(-1)`
Promotional Banner

Topper's Solved these Questions

  • MATRICES

    CENGAGE|Exercise Exercise 13.1|5 Videos

  • MATRICES

    CENGAGE|Exercise Exercise 13.2|6 Videos

  • MATHMETICAL REASONING

    CENGAGE|Exercise JEE Previous Year|10 Videos

  • METHODS OF DIFFERENTIATION

    CENGAGE|Exercise Single Correct Answer Type|46 Videos

Similar Questions

Explore conceptually related problems

If A and B are non-singular matrices, then

Statement 1: If the matrices,A,B,(A+B) are non-singular,then [A(A+B)^(-1)B]^(-1)=B^(-1)+A^(-1). Statement 2:[A(A+B)^(-1)B]^(-1)=[A(A^(-1)+B^(-1))B]^(-1)=[(I+AB^(-1))B]^(-1)=[(B+AB^(-1))B]^(-1)=[(B+AI)]^(-1)=[(B+A)]^(-1)=B^(-1)+A^(-1)

If A and B be two non singular matrices and A^(-1) and B^(-1) are their respective inverse, then prove that (AB)^(-1)=B^(-1)A^(-1)

If A and B are non-singular matrices such that B^(-1)AB=A^(3), then B^(-3)AB^(3)=

If the two matrices A,B,(A+B) are non-singular (where A and B are of the same order),then (A(A+B)^(-1)B)^(-1) is equal to (A)A+B(B)A^(-1)+B^(-1)(C)(A+B)^(-1)(D)AB

If A and B are two non-singular matrices which commute, then (A(A+B)^(-1)B)^(-1)(AB)=

If A and B are two non-singular matrices which commute, then (A(A+B)^(-1)B)^(-1)(AB)=

Prove that: :(a^(-1)+b^(-1))^(-1)=(ab)/(a+b)

If A,B are two n xx n non-singular matrices, then

CENGAGE-MATRICES-Solved Examples And Exercises
  1. If the matrices, A, B and (A+B) are non-singular, then prove that [A(A...

    Text Solution

    |

  2. In which of the following type of matrix inverse does not exist always...

    Text Solution

    |

  3. If both A-1/2Ia n dA+1/2 are orthogonal matices, then (a)A is ortho...

    Text Solution

    |

  4. If nth-order square matrix A is a orthogonal, then |adj(adjA)| is (...

    Text Solution

    |

  5. If P is an orthogonal matrix and Q=P A P^T an dx=P^T Q^1000 P then x^...

    Text Solution

    |

  6. If A is a nilpotent matrix of index 2, then for any positive integer n...

    Text Solution

    |

  7. If Aa n dB are two matrices such that A B=Ba n dB A=A ,t h e n (A^5-B...

    Text Solution

    |

  8. If Z is an idempotent matrix, then (I+Z)^n I+2^n Z b. I+(2^n-1)Z c. ...

    Text Solution

    |

  9. If A is an orthogonal matrix then A^(-1) equals A^T b. A c. A^2 d. non...

    Text Solution

    |

  10. If A^2=1, then the value of det(A-I) is (where A has order 3) 1 b. -1 ...

    Text Solution

    |

  11. Let A be an nth-order square matrix and B be its adjoint, then |A B+K ...

    Text Solution

    |

  12. A=[(a,1,0),(1,b,d),(1,b,c)],B=[(a,1,1),(0,d,c),(f,g,h)],U=[(f),(g),(h)...

    Text Solution

    |

  13. If M is a 3 xx 3 matrix, where det M=1 and MM^T=1, where I is an ident...

    Text Solution

    |

  14. If A is a diagonal matrix of order 3xx3 is commutative with every squa...

    Text Solution

    |

  15. Let S be the set which contains all possible vaues fo I ,m ,n ,p ,q ,r...

    Text Solution

    |

  16. Given a matrix A=[a b c b c a c a b],w h e r ea ,b ,c are real positiv...

    Text Solution

    |

  17. If A is a square matrix of order 3 such that |A|=2,t h e n|(a d jA^(-1...

    Text Solution

    |

  18. Let A=[[3x^2], [1], [6x]],B=[a,b,c]and C=[[(x+2)^2, 5x^2, 2x],[5x^2, 2...

    Text Solution

    |

  19. If A is an idempotent matrix satisfying, (I-0. 4 A)^(-1)=I-alphaA ,w h...

    Text Solution

    |

  20. Let A=([a(i j)])(3xx3) be a matrix such that AA^T=4Ia n da(i j)+2c(i j...

    Text Solution

    |

  21. Let A be the set of all 3xx3 skew-symmetri matrices whose entries are ...

    Text Solution

    |