Home
Class 12
MATHS
If B is an idempotent matrix, and A=I-B ...

If `B` is an idempotent matrix, and `A=I-B ,` then `A^2=A` b. `A^2=I` c. `A B=O` d. `B A=O`

A

`A^(2)=A`

B

`A^(2)=I`

C

`AB=O`

D

`BA=O`

Text Solution

Verified by Experts

The correct Answer is:
A, C, D
B is an idempotent matrix
`:. B^(2)=B`
Now, `A^(2)=(I-B)^(2)`
`=(I-B) (I-B)`
`=I-IB-IB+B^(2)`
`=I-B-B+B^(2)`
`=I-2B+B^(2)`
`=I-2B+B`
`=I-B`
`=A`
Therefore, A is idempotent. Again,
`AB=(I-B)B=IB-B^(2)=B-B^(2)=B^(2)=B^(2)-B^(2)=O`
Similarly, `BA=B(I-B)=BI-B^(2)=B-B=O`.
Promotional Banner

Topper's Solved these Questions

  • MATHMETICAL REASONING

    CENGAGE PUBLICATION|Exercise Archives|10 Videos

  • METHODS OF DIFFERENTIATION

    CENGAGE PUBLICATION|Exercise Multiple Correct Answer Type|7 Videos

Similar Questions

Explore conceptually related problems

If B is an idempotent matrix, and A=I-B , then a. A^2=A b. A^2=I c. A B=O d. B A=O

If A B=Aa n dB A=B , then a. A^2B=A^2 b. B^2A=B^2 c. A B A=A d. B A B=B

If Z is an idempotent matrix, then (I+Z)^n I+2^n Z b. I+(2^n-1)Z c. I-(2^n-1)Z d. none of these

If A ,B ,A+I ,A+B are idempotent matrices, then A B is equal to a. B A b. -B A c. I d. O

If [[a, b],[ c,1-a]] is an idempotent matrix and f(x)=x-x^2 , bc=1/4 , then the value of 1//f(a) is ______.

A square matrix A is used to be an idempotent matrix if A^(2) = A . If A, B and A + B are idempotent matrices, then-

A square matrix A is used to be an idempotent matrix if A^(2) = A . If A is an idempotent matrix and B = I -A, then-

If A is non-diagonal involuntary matrix, then a. A-I=O b. A+I=O c. A-I is nonzero singular d. none of these

If A is a nilpotent matrix of index 2, then for any positive integer n ,A(I+A)^n is equal to (a) A^(-1) (b) A (c) A^n (d) I_n

If A and B are square matrices of order 3 such that A^(3)=8 B^(3)=8I and det. (AB-A-2B+2I) ne 0 , then identify the correct statement(s), where I is identity matrix of order 3. (A) A^(2)+2A+4I=O (B) A^(2)+2A+4I neO (C) B^(2)+B+I=O (D) B^(2)+B+I ne O

CENGAGE PUBLICATION-MATRICES-All Questions
  1. If A(theta)=[(sin theta, i cos theta),(i cos theta, sin theta)], then ...

    Text Solution

    |

  2. Let Aa n dB be two nonsingular square matrices, A^T a n dB^T are the t...

    Text Solution

    |

  3. If B is an idempotent matrix, and A=I-B , then A^2=A b. A^2=I c. A B=O...

    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

    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 |A|=-1 b. adj A=[[-...

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

  20. If Aa n dB are three square matrices of the same order, then A B=A C ...

    Text Solution

    |