Home
Class 12
MATHS
If A1=[(0, 0, 0, 1), (0, 0, 1, 0), (0, 1...

If `A_1=[(0, 0, 0, 1), (0, 0, 1, 0), (0, 1, 0, 0), (1, 0, 0, 0)],A_2=[(0, 0, 0,i),(0, 0,-i,0),( 0,i,0, 0),(-i,0, 0, 0)],t h e nA_i A_k+A_k A_i` is equal to `2lifi=k` b. `Oifi!=k` c. `2lifi!=k` d. `O` always

A

`2I` if `i=k`

B

`O` if `i ne k`

C

`2I` if `i ne k`

D

`O` always

Text Solution

Verified by Experts

The correct Answer is:
A, B
Let `i=k=1` (say). Then,
`A_(r)A_(k)=A_(k)A_(i)=A_(1)A_(1)`
`A_(i)A_(k)=A_(1)A_(1)=[(0,0,0,0),(0,0,1,0),(0,1,0,0),(1,0,0,0)]xx[(0,0,0,1),(0,0,1,0),(0,1,0,0),(1,0,0,0)]`
`=[(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)]=I`
`A_(2)A_(2)=[(0,0,0,i),(0,0,-i,0),(0,i,0,0),(-i,0,0,0)]xx[(0,0,0,i),(0,0,-i,0),(0,i,0,0),(-i,0,0,0)]`
`=[(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)]=I`
`:. A_(i)A_(k)+A_(k)A_(i)=I+I=2I`
If `i ne k` let `i=1` and `k=2`, then
`A_(i)A_(k)=A_(1)A_(2)=[(0,0,0,1),(0,0,1,0),(0,1,0,0),(1,0,0,0)]xx[(0,0,0,i),(0,0,-i,0),(0,i,0,0),(-i,0,0,0)]`
`=[(-i,0,0,0),(0,i,0,0),(0,0,-i,0),(0,0,0,i)]`
Also, `A_(2)A_(1)=[(0,0,0,i),(0,0,-i,0),(0,i,0,0),(-i,0,0,0)]xx[(0,0,0,1),(0,0,1,0),(0,1,0,0),(1,0,0,0)]`
`=[(i,0,0,0),(0,-i,0,0),(0,0,i,0),(0,0,0,-i)]`
`implies A_(1)A_(2)+A_(2)A_(1)=O`
Promotional Banner

Topper's Solved these Questions

  • MATRICES

    CENGAGE ENGLISH|Exercise Linked Comprehension Type|27 Videos

  • MATRICES

    CENGAGE ENGLISH|Exercise Matrix Type|5 Videos

  • MATRICES

    CENGAGE ENGLISH|Exercise Exercises|65 Videos

  • MATHMETICAL REASONING

    CENGAGE ENGLISH|Exercise Archives|10 Videos

  • METHODS OF DIFFERENTIATION

    CENGAGE ENGLISH|Exercise Single Correct Answer Type|46 Videos

Similar Questions

Explore conceptually related problems

If A=[(1,0,0),(0,-1,0),(0,0,2)] and B=[(2,0,0),(0, 3,0),(0,0,-1)] , find A+B , 3A+4B .

If A=[(i,0), (0,i)],\ n in N , then A^(4n) equals [(0,i), (i,0)] (b) [(0 ,0) ,(0, 0)] (c) [(1, 0) ,(0, 1)] (d) [(0,i) ,(i,0)]

If A=[(0,0,0,0),(0,0,0,0),(1,0,0,0),(0,1,0,0)] then (A) A^2=I (B) A^2=0 (C) A^3=0 (D) none of these

If A=[(3, 2, 0),( 1, 4 ,0 ),(0 ,0 ,5)] , show that A^2-7A+10 I=O .

If A=[{:(1,0,0),(0,1,0),(a,b,-1):}] then A^(2) is equal to

If A=[{:(1,0,0),(0,1,0),(a,b,-1):}] then A^(2) is equal to

If A=[[i,0], [0,i]] , write A^2 .

If A=[(2, 3),( 1, 2)] , verify that A^2-4A+I=O , where I=[(1, 0),( 0, 1)] and O=[(0, 0),( 0, 0)] . Hence, find A^(-1) .

If A=|{:(1,0,0),(0,1,0),(0,0,1):}|" and A"=[{:(0,-3,4),(1,2,3),(0,5,5):}]," then find "(I-A)^(-1)

A= [{:( 1,0,0) ,( 0,1,1) , ( 0,-2,4) :}] ,I= [{:( 1,0,0) ,( 0,1,0),( 0,0,1) :}]and A^(-1) =[(1)/(6) (A^(2)+cA +dt)] then , the value of c and d are

CENGAGE ENGLISH-MATRICES-Multiple Correct Answer
  1. Let A and B be two nonsingular square matrices, A^(T) and B^(T) are th...

    Text Solution

    |

  2. If B is an idempotent matrix, and A=I-B, then

    Text Solution

    |

  3. If A1=[(0, 0, 0, 1), (0, 0, 1, 0), (0, 1, 0, 0), (1, 0, 0, 0)],A2=[(0,...

    Text Solution

    |

  4. Suppose a(1), a(2), .... Are real numbers, with a(1) ne 0. If a(1), a(...

    Text Solution

    |

  5. If alpha,beta,gamma are three real numbers and A=[(1,"cos"(alpha-beta)...

    Text Solution

    |

  6. If D(1) and D(2) are two 3xx3 diagonal matrices, then which of the fol...

    Text Solution

    |

  7. Let A be the 2 xx 2 matrix given by A=[a("ij")] where a("ij") in {0, 1...

    Text Solution

    |

  8. If S=[(0,1,1),(1,0,1),(1,1,0)]a n dA=[(b+c,c+a,b-c),(c-b,c+b, a-b),(b-...

    Text Solution

    |

  9. P is a non-singular matrix and A, B are two matrices such that B=P^(-1...

    Text Solution

    |

  10. Let A=[(1,2,2),(2,1,2),(2,2,1)] . Then A^2-4A-5I3=O b. A^(-1)=1/5(A-4...

    Text Solution

    |

  11. If A=[{:(1,0,0),(1,0,1),(0,1,0):}], then which is true a. A^(3)-A^(2)=...

    Text Solution

    |

  12. If Ais symmetric and B is skew-symmetric matrix, then which of the fol...

    Text Solution

    |

  13. If A=((a(i j)))(nxxn) and f is a function, we define f(A)=((f(a(i j)))...

    Text Solution

    |

  14. If A is a matrix such that A^2+A+2I=Odot, the which of the following i...

    Text Solution

    |

  15. If A=[{:(3,-3,4),(2,-3,4),(0,-1,1):}] , then the trace of the matrix A...

    Text Solution

    |

  16. If [(1,-tan theta),(tan theta,1)][(1,tan theta),(-tan theta,1)]^(-1)=[...

    Text Solution

    |

  17. If A^(-1)=[(1,-1,2),(0,3,1),(0,0,-1//3)] , then :

    Text Solution

    |

  18. If A is an invertible matrix, then (a d jdotA)^(-1) is equal to a.a d...

    Text Solution

    |

  19. If A and B are two invertible matrices of the same order, then adj (AB...

    Text Solution

    |

  20. If A, B, and C are three square matrices of the same order, then AB=AC...

    Text Solution

    |