Home
Class 12
MATHS
If A=[1tanx-tanx1], show that A^T A^(-1)...

If `A=[1tanx-tanx1],` show that `A^T A^(-1)=[cos2x-sin2xsin2xcos2x]`

A

`[(-"cos" 2x,"sin" 2x),(-sin 2x,cos 2x)]`

B

`[(cos 2x,-sin 2x),(sin 2x, cos 2x)]`

C

`[(cos 2x,cos 2x),(cos 2x, sin 2x)]`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
`|A|=|(1,tan x),(-tan x,1)|=1+tan^(2) x ne 0`
So A is invertiable. Also,
adj `A=[(1,tan x),(-tan x,1)]^(T)=[(1,-tan x),(tan x,1)]`
Now, `A^(-1)=1/(|A|)` adj A
`= 1/((1+tan^(2) x)) [(1,-tan x),(tan x,1)]`
`=[(1/(1+tan^(2) x),(-tan x)/(1+tan^(2) x)),((tan x)/(1+tan^(2) x),1/(1+tan^(2)x))]`
`:. A^(T) A^(-1)=[(1,-tan x),(tan x,1)][(1/(1+tan^(2) x),(-tan x)/(1+tan^(2) x)),((tan x)/(1+tan^(2) x),1/(1+tan^(2)x))]`
`=[((1-tan^(2) x)/(1+tan^(2) x),(-2 tan x)/(1+tan^(2)x)),((2 tan x)/(1+tan^(2)x),(1-tan^(2) x)/(1+tan^(2) x))]`
`=[(cos 2x,-sin 2 x),(sin 2x,cos 2x)]`
Promotional Banner

Topper's Solved these Questions

  • MATRICES

    CENGAGE|Exercise Exercise (Multiple)|33 Videos

  • MATRICES

    CENGAGE|Exercise Exercise (Comprehension)|18 Videos

  • MATRICES

    CENGAGE|Exercise Exercise 13.5|17 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=[1tan x-tan x1], show that A^(T)A^(-1)=[cos2x-sin2x sin2x cos2x]

If A=[[1,tan x-tan x,1]], show that A^(T)A^(-1)=[[cos2x,-sin2xsin2x,cos2x]]

If A=[(1,tanx),(-tanx, 1)], " show that " A' A^(-1)=[(cos 2x,-sin 2x),(sin 2x, cos 2x)] .

If y = {(1-tanx)/(1+tanx)} , show that (dy)/(dx) = (-2)/((1+sin2x)) .

If y=(1-tanx)/(1+tanx) , prove that (dy)/(dx)=(-2)/(1+sin2x) .

int(sin^8x-cos^8x)/(1-2sin^2xcos^2x)dx

If A= [{:( 1,-tanx ),( tanx ,1):}] then the value of [A^(T) A^(-1) ] is

(1)/(sin x.cos^(2)x)

int(2tanx+3)/(sin^2x+2cos^2x)dx

Let P(x)=cot^(2)x ((1+tanx+tan^(2)x)/(1+cot x+ cot^(2)x))+((cos x-cos 3x+sin3x-sin x)/(2(sin 2x+cos2x)))^(2) . Then, which of the following is (are) correct ?

CENGAGE-MATRICES-Exercise (Single)
  1. There are two possible values of A in the solution of the matrix equat...

    Text Solution

    |

  2. If A and B are two square matrices such that B=-A^(-1)BA, then (A+B)^(...

    Text Solution

    |

  3. If A=[1tanx-tanx1], show that A^T A^(-1)=[cos2x-sin2xsin2xcos2x]

    Text Solution

    |

  4. If A is order 3 square matrix such that |A|=2, then |"adj (adj (adj A)...

    Text Solution

    |

  5. If A=[(0,1,2),(1,2,3),(3,a,1)] and A^(-1)=[(1//2,-1//2,1//2),(-4,3,c),...

    Text Solution

    |

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

    Text Solution

    |

  7. Let aa n db be two real numbers such that a >1,b > 1. If A=(a0 0b) , t...

    Text Solution

    |

  8. If A=[a("ij")](4xx4), such that a("ij")={(2",","when "i=j),(0",","when...

    Text Solution

    |

  9. A is an involuntary matrix given by A=[0 1-1 4-3 4 3-3 4] , then the i...

    Text Solution

    |

  10. If A is a nonsingular matrix such that A A^(T)=A^(T)A and B=A^(-1) A^(...

    Text Solution

    |

  11. If P is an orthogonal matrix and Q=P A P^T an dx=P^T A b. I c. A^(100...

    Text Solution

    |

  12. If Aa n dB are two non-singular matrices of the same order such that B...

    Text Solution

    |

  13. If adjB=A ,|P|=|Q|=1,then adj(Q^(-1)B P^(-1)) is

    Text Solution

    |

  14. If A is non-singular and (A-2I)(A-4I)=O ,t h e n1/6A+4/3A^(-1) is equa...

    Text Solution

    |

  15. Let f(x)=(1+x)/(1-x) . If A is matrix for which A^3=O ,t h e nf(A) is ...

    Text Solution

    |

  16. if[{:(2,1),(3,2):}]A[{:(-3,2),(5,-3):}]=[{:(1,0),(0,1):}],"then" A=?

    Text Solution

    |

  17. If A^2-A +I = 0, then the inverse of A is: (A) A+I (B) A (C) ...

    Text Solution

    |

  18. If F(x)=[("cos"x,-sin x,0),(sin x,cos x,0),(0,0,1)] and G(y)=[(cos y,0...

    Text Solution

    |

  19. If Aa n dB are square matrices of the same order and A is non-singular...

    Text Solution

    |

  20. If k in Rot h e ndet{a d j(k In)} is equal to K^(n-1) b. K^(n(n-1)) c...

    Text Solution

    |