Home
Class 12
MATHS
If Aa n dB are two nonzero square matric...

If `Aa n dB` are two nonzero square matrices of the same ordr such that the product `A B=O ,` then both A and B must be singular exactly one of them must be singular both of them are non singular none of these

A

both A and B must be singular

B

exactly one of them must be singular

C

both of them are nonsingular

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
A
If possible assume that A is nonsingular, then `A^(-1)` exists. Thus,
`AB=O implies A^(-1) (AB)=(A^(-1) A)B=O`
`implies IB=O` or `B=O` (contradiction)
Hence, both A and B must be singular.
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 A and B are two nonzero square matrices of the same order such that the product AB=O , then

If A and B are two non singular matrices and both are symmetric and commute each other, then

A and B are non-singular matrices of same order then show that adj(AB) = (adjB) (adjA)

Let A and B be two square matrices of the same size such that AB^(T)+BA^(T)=O . If A is a skew-symmetric matrix then BA is

If A and B are two non-singular matrices of order 3 such that A A^(T)=2I and A^(-1)=A^(T)-A . Adj. (2B^(-1)) , then det. (B) is equal to

If A ,\ B are square matrices of order 3,\ A is non-singular and A B=O , then B is a (a) null matrix (b) singular matrix (c) unit matrix (d) non-singular matrix

If Aa n dB are two non-singular matrices of the same order such that B^r=I , for some positive integer r >1,t h e nA^(-1)B^(r-1)A-A^(-1)B^(-1)A= a. I b. 2I c. O d. -I

Statement 1: Let A ,B be two square matrices of the same order such that A B=B A ,A^m=O ,n dB^n=O for some positive integers m ,n , then there exists a positive integer r such that (A+B)^r=Odot Statement 2: If A B=B A then (A+B)^r can be expanded as binomial expansion.

If A and B are two non-singular matrices which commute, then (A(A+B)^(-1)B)^(-1)(AB)=

If A and B are square matrices of order n , then prove that Aa n dB will commute iff A-lambdaIa n dB-lambdaI commute for every scalar lambdadot

CENGAGE PUBLICATION-MATRICES-All Questions
  1. If matrix A is given by A=|[6, 11], [2, 4]| , then the determinant o...

    Text Solution

    |

  2. If A is non-diagonal involuntary matrix, then a. A-I=O b. A+I=O c. A-...

    Text Solution

    |

  3. If Aa n dB are two nonzero square matrices of the same ordr such that ...

    Text Solution

    |

  4. If Aa n dB are symmetric matrices of the same order and X=A B+B Aa n d...

    Text Solution

    |

  5. If A ,B ,A+I ,A+B are idempotent matrices, then A B is equal to a.B A ...

    Text Solution

    |

  6. If A=[(0,x),(y,0)] and A^(3)+A=O then sum of possible values of xy is

    Text Solution

    |

  7. Which of the following is an orthogonal matrix ? ( ⎡ ⎢ ⎣ 6 / 7 2 / 7...

    Text Solution

    |

  8. Let A and B be two square matrices of the same size such that AB^(T)+B...

    Text Solution

    |

  9. In which of the following type of matrix inverse does not exist always...

    Text Solution

    |

  10. Let A be an nth-order square matrix and B be its adjoint, then |A B+K ...

    Text Solution

    |

  11. If A=[(a, b, c), (x, y, z), (p, q, r)], B=[(q,-b, y),(-p, a,-x),(r,-c,...

    Text Solution

    |

  12. If A(alpha,beta)=[cosalphas inalpha0-s inalphacosalpha0 0 0e^(beta)],t...

    Text Solution

    |

  13. If A=[[a+i b, c+i d],[-c+i d, a-i b]] and a^2+b^2+c^2+d^2=1,t h e n ...

    Text Solution

    |

  14. Id [(1/25, 0), (x,1/25)]=[(5, 0), (-a, 5)]^(-2) , then the value of x ...

    Text Solution

    |

  15. If A = [[1 ,2],[2,1]]and f(x)=(1+x)/(1-x), then f(A) is

    Text Solution

    |

  16. There are two possible values of A in the solution of the matrix equat...

    Text Solution

    |

  17. If A and B are two square matrices such that B=-A^(-1)B A ,t h e n(A+B...

    Text Solution

    |

  18. If A=[(1,tanx),(-tanx,1)] , show that A^T\ A^(-1)=[(cos2x,-sin2x),(sin...

    Text Solution

    |

  19. If A is order 2 square matrix such that |A|=2, then |(adj(adj(adjA)))|...

    Text Solution

    |

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

    Text Solution

    |