Home
Class 12
MATHS
Let A=[(2,1),(0,3)] be a matrix. If A^(1...

Let `A=[(2,1),(0,3)]` be a matrix. If `A^(10)=[(a,b),(c,d)]` then prove that `a+d` is divisible by 13.

Text Solution

Verified by Experts

We have
`A^(2)=[(2,1),(0,3)][(2,1),(0,3)]=[(4,5),(0,9)]`
`A^(3)=A^(2)A=[(4,5),(0,9)][(2,1),(0,3)]=[(8,19),(0,27)]`
`implies A^(n) =[(2^(n),3^(n)-2^(n)),(0, 3^(n))]`
Now `A^(10)=[(a,b),(c,d)]`
`implies a=2^(10), d=3^(10)`
So, `a+b=2^(10)+3^(10)=4^(5)+9^(5)`, which is multiple of 13.
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

Let A=[(0, alpha),(0,0)] and (A+I)^(50) -50A=[(a,b),(c,d)] . Then the value of a+b+c+d is

If the points (a, 0), (b,0), (0, c) , and (0, d) are concyclic (a, b, c, d > 0) , then prove that ab = cd .

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 b^2<2a c , then prove that a x^3+b x^2+c x+d=0 has exactly one real root.

Let a and b be two real numbers such that a > 1, b > 1. If A=[(a,0), (0,b)] , then lim_(n to oo) A^(-n) is a. unit matrix b. null matrix c. 2l d. none of these

Let A=[[1, 2], [3 , 4]] and B = [[a, b],[ c, d]] be two matrices such that they are commutative and c ne 3 b then the value of |(a-d)/(2 b-c)| is

Let X = { 1,2,3,4}, Y = {a,b,c,d} and f={(1,a), (4,b), (2,c),(3,d),(2,d)}. Then f is

Let f(x)=(ax + b )/(cx+d) . Then the fof (x)=x , provided that : (a!=0, b!= 0, c!=0,d!=0)

A=[(a,1,0),(1,b,d),(1,b,c)],B=[(a,1,1),(0,d,c),(f,g,h)],U=[(f),(g),(h)],V=[(a^2),(0),(0)] If there is a vector matrix X, such that AX = U has infinitely many solutions, then prove that BX = V cannot have a unique solution. If a f d != 0 . Then,prove that BX = V has no solution.

CENGAGE-MATRICES-Examples
  1. If A=[costhetasintheta-sinthetacostheta], then prove that A^n=[cosnthe...

    Text Solution

    |

  2. If A=((p,q),(0,1)), then show that A^(8)=((p^(8),q((p^(8)-1)/(p-1))),(...

    Text Solution

    |

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

    Text Solution

    |

  4. Show that the solution of the equation [(x, y),(z, t)]^(2)=O is [(x,y)...

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

  8. Let A=[(1,2),(-1,3)] .If A^6=kA-205I then then numerical quantity of...

    Text Solution

    |

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

    Text Solution

    |

  10. If A and B are square matrices of the same order such that AB = BA, th...

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

  14. 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

    |

  15. 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

    |

  16. 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

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |