Home
Class 12
MATHS
If A and B are square matrices of same o...

If A and B are square matrices of same order such that `AB=O` and `B ne O`, then prove that `|A|=0`.

Text Solution

Verified by Experts

We have `AB=O`
`:. |AB|=0`
`implies |A||B|=0`
`implies |A|=0` or `|B|=0`
Now, let `|A| ne 0`, then `A^(-1)` exists.
Thus, from `AB=O`,
`A^(-1) AB=A^(-1)O`
`:. IB=O`
or `B=O`
But it is given that `B ne O`, then `|A|=0`.
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 square matrices of the same order such that AB = A and BA = B, then……

If A and B are square matrices of same order such that AB+BA=O , then prove that A^(3)-B^(3)=(A+B) (A^(2)-AB-B^(2)) .

If A and B are square matrices of the same order such that A B = B A , then prove by induction that A B^n=B^n A .

If A and B are square matrices of the same order such that AB = BA, then prove by induction that AB^(n)=B^(n)A . Further, prove that (AB)^(n)=A^(n)B^(n) for all n in N .

If A and B are square matrices of the same order, then AB = 0 implies

If A and B are square matrices of order 3 such that |A| = 3 and |B| = 2 , then find the value of |A^(-1) adj(B^(-1)) adj (2A^(-1))|

If A and B are square matrices of the same order such that A^(2)=A,B^(2)=B,AB=BA=0 , then__

Let A and B be square matrices of the same order such that A^(2)=I and B^(2)=I , then which of the following is CORRECT ?

If A and B are two square matrices of the same order such that AB = A, BA = B and if a matrix A is called idempotent if A^(2)=A , then

If A and B are two nonzero square matrices of the same order such that the product AB=O , then

CENGAGE PUBLICATION-MATRICES-All Questions
  1. If M is the matrix [(1,-3),(-1,1)] then find matrix sum(r=0)^(oo) ((-1...

    Text Solution

    |

  2. Let p be a non singular matrix, and I + P + p^2 + ... + p^n = 0, then ...

    Text Solution

    |

  3. If A and B are square matrices of same order such that AB=O and B ne O...

    Text Solution

    |

  4. If A is a symmetric matrix, B is a skew-symmetric matrix, A+B is nonsi...

    Text Solution

    |

  5. If the matrices, A ,B ,(A+B) are non-singular, then prove that [A(A+B)...

    Text Solution

    |

  6. If matrix a satisfies the equation A^(2)=A^(-1), then prove that A^(2^...

    Text Solution

    |

  7. If A and B are non-singular symmetric matrices such that AB=BA, then p...

    Text Solution

    |

  8. If A is a matrix of order n such that A^(T)A=I and X is any matrix suc...

    Text Solution

    |

  9. Show that two matrices A=[(1,-1,0),(2,1,1)] and B=[(3,0,1),(0,3,1)] ...

    Text Solution

    |

  10. Using elementary transformations, find the inverse of the matrix : ...

    Text Solution

    |

  11. Let A be a 3xx3 matric such that A . [(1,2,3),(0,2,3),(0,1,1)]=[(0,0,...

    Text Solution

    |

  12. Solve the following system of equations, using matrix method. x+2y+z=7...

    Text Solution

    |

  13. Using matrix method, show that following system of equation is inconsi...

    Text Solution

    |

  14. If f(x) and g(x) are two functions with g(x)=x−1/x and fog(x) =x^3−1/ ...

    Text Solution

    |

  15. Find the characteristic roots of the two-rowed orthogonal matrix [(cos...

    Text Solution

    |

  16. Show that if lambda(1), lambda(2), ...., lambda(n) are n eigenvalues o...

    Text Solution

    |

  17. If A is nonsingular, prove that the eigenvalues of A^(-1) are the reci...

    Text Solution

    |

  18. If one of the eigenvalues of a square matrix a order 3xx3 is zero, the...

    Text Solution

    |

  19. Construct a 3 xx 4 matrix, whose elements are given by: a(i j)=1/2|-3...

    Text Solution

    |

  20. Find the value of a if [a-b2a+c2a-b3c+d]=[-1 5 0 13]

    Text Solution

    |