Home
Class 12
MATHS
If A^(3)=O, then prove that (I-A)^(-1) =...

If `A^(3)=O`, then prove that `(I-A)^(-1) =I+A+A^(2)`.

Text Solution

Verified by Experts

Given`A^(3)=O`
Now, `(I-A) (I+A+A^(2))`
`=I^(2)+IA+IA^(2)-AI-A^(2)-A^(3)`
`=I+A+A^(2)-A-A^(2)-O`
`=I`
`:. (I-A)^(-1)=I+A+A^(2)`
Promotional Banner

Topper's Solved these Questions

  • MATRICES

    CENGAGE|Exercise Exercise (Single)|65 Videos

  • MATRICES

    CENGAGE|Exercise Exercise (Multiple)|33 Videos

  • MATRICES

    CENGAGE|Exercise Exercise 13.4|12 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 is a 2xx2 matrix such that A^(2)-4A+3I=O , then prove that (A+3I)^(-1)=7/24 I-1/24 A .

If A is a square matrix such that A^(3)=I , then prove that A is non-singular

If A=[(a,b),(c,d)] , where a, b, c and d are real numbers, then prove that A^(2)-(a+d)A+(ad-bc) I=O . Hence or therwise, prove that if A^(3)=O then A^(2)=O

If A^3=O ,t h e nI+A+A^2 equals a. I-A b. (I+A^1)^(-1) c. (I-A)^(-1) d. none of these

If A=[(1,0,2),(0,2,1),(2,0,3)] , prove that A^(3)-6A^(2)+7A+2I=0

A triangle has vertices A_(i) (x_(i),y_(i)) for i= 1,2,3,. If the orthocenter of triangle is (0,0) then prove that |{:(x_(2)-x_(3),,y_(2)-y_(3),,y_(1)(y_(2)-y_(3))+x_(1)(x_(2)-x_(3))),(x_(3)-x_(1) ,,y_(3)-y_(1),,y_(2)(y_(3)-y_(1))+x_(2)(x_(3)-x_(1))),( x_(1)-x_(2),,y_(1)-y_(2),,y_(3)(y_(1)-y_(2))+x_(3)(x_(1)-x_(2))):}|=0

Let f(x)=(1+x)/(1-x) . If A is matrix for which A^3=O ,t h e nf(A) is (a) I+A+A^2 (b) I+2A+2A^2 (c) I-A-A^2 (d) none of these

If z=(sqrt(3)/2+i/2)^5+(sqrt(3)/2-i/2)^5 , then prove that Im(z)=0

" if " x_(i) =a_(i) b_(i) C_(i), i= 1,2,3 are three- digit positive integer such that each x_(i) is a mulptiple of 19 then prove that det {{:(a_(1),,a_(2),,a_(3)),(b_(1),,b_(2),,b_(3)),(c_(1),,c_(2),,c_(3)):}} is divisible by 19.

If A=[[5,3],[-1,-2]] , show that A^(2)-3A-7I_(2)=O_(2) Hence find A^(-1) .

CENGAGE-MATRICES-Exercise 13.5
  1. By the method of matrix inversion, solve the system. [(1,1,1),(2,5,7...

    Text Solution

    |

  2. Let A=[[2,0,7] , [0,1,0], [1,-2,1]] and B=[[-x,14x,7x] , [0,1,0] , [x,...

    Text Solution

    |

  3. If A=[{:(0,1,1),(1,0,1),(1,1,0):}] show that A^(-1)=(1)/(2)(A^(2)=3I)

    Text Solution

    |

  4. For the matrix A=[3 1 7 5] , find x and y so that A^2+x I=y Adot

    Text Solution

    |

  5. If A^(3)=O, then prove that (I-A)^(-1) =I+A+A^(2).

    Text Solution

    |

  6. If A=[[cos alpha, -sin alpha] , [sin alpha, cos alpha]], B=[[cos2beta,...

    Text Solution

    |

  7. If A=[(1,2,2),(2,2,3),(1,-1,3)], C=[(2,1,1),(2,2,1),(1,1,1)], D=[(10),...

    Text Solution

    |

  8. If A is a 2xx2 matrix such that A^(2)-4A+3I=O, then prove that (A+3I)^...

    Text Solution

    |

  9. For two unimobular complex numbers z(1) and z(2), find [(bar(z)(1),-z(...

    Text Solution

    |

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

    Text Solution

    |

  11. If square matrix a is orthogonal, then prove that its inverse is also ...

    Text Solution

    |

  12. If A is a skew symmetric matrix, then B=(I-A)(I+A)^(-1) is (where I is...

    Text Solution

    |

  13. Prove that ("adj. "A)^(-1)=("adj. "A^(-1)).

    Text Solution

    |

  14. Using elementary transformation, find the inverse of the matrix A=[(a,...

    Text Solution

    |

  15. Show that the two matrices A, P^(-1) AP have the same characteristic r...

    Text Solution

    |

  16. Show that the characteristics roots of an idempotent matris are either...

    Text Solution

    |

  17. If alpha is a characteristic root of a nonsin-gular matrix, then prove...

    Text Solution

    |