Home
Class 12
MATHS
For the following matrices verify the ...

For the following matrices verify the associativity of matrix multiplication i.e. `(A B)C=A(B C)`
`(i) A=[(1 ,2 ,0),(-1 ,0, 1)]` , `B=[(1 ,0),(-1 ,2),( 0 ,3)]` and `C=[(1),(-1)]`
(ii) `A=[(4, 2 ,3),( 1 ,1 ,2 ),(3 ,0 ,1)]` , `B=[(1,-1 ,1),( 0 ,1, 2),( 2,-1, 1)]` and `C=[(1 ,2,-1 ),(3 ,0 ,1),( 0 ,0 ,1)]` .

Text Solution

Verified by Experts

`(i)Given,
A=[(1 ,2 ,0),(-1 ,0, 1)]`
`B=[(1 ,0),(-1 ,2),( 0 ,3)]`
and `C=[(1),(-1)]`
`(AB)C=([(1 ,2 ,0),(-1 ,0, 1)][(1 ,0),(-1 ,2),( 0 ,3)])[(1),(-1)]`
`=[(1-2+0,0+4+0),(-1+0+0,0+0+3)][(1),(-1)]`
`=[(-1,4),(-1,3)][(1),(-1)]`
`=[(-1-4),(-1-3)]`
`=[(-5),(-4)]`
`A(BC)=[(1 ,2 ,0),(-1 ,0, 1)]([(1 ,0),(-1 ,2),( 0 ,3)][(1),(-1)])`
`=[(1 ,2 ,0),(-1 ,0, 1)][(1+0),(-1-2),(0-3)]`
`=[(1 ,2 ,0),(-1 ,0, 1)][(1),(-3),(-3)]`
`=[(-5),(-4)]`
Clearly `(AB) C = A (BC) `
`(ii)`Given,
`A=[(4, 2 ,3),( 1 ,1 ,2 ),(3 ,0 ,1)]`
`B=[(1,-1 ,1),( 0 ,1, 2),( 2,-1, 1)]`
and `C=[(1 ,2,-1 ),(3 ,0 ,1),( 0 ,0 ,1)]`
`(AB)C=([(4, 2 ,3),( 1 ,1 ,2 ),(3 ,0 ,1)][(1,-1 ,1),( 0 ,1, 2),( 2,-1, 1)])[(1 ,2,-1 ),(3 ,0 ,1),( 0 ,0 ,1)]`
`=[(10 ,-5,11 ),(5 ,-2 ,5),( 5,-4,4)][(1 ,2,-1 ),(3 ,0 ,1),( 0 ,0 ,1)]`
`=[(7,20,-22),(-10,10,-1),(-5,10,5)]`
again,
`A(BC)=[(4, 2 ,3),( 1 ,1 ,2 ),(3 ,0 ,1)]([(1,-1 ,1),( 0 ,1, 2),( 2,-1, 1)][(1 ,2,-1 ),(3 ,0 ,1),( 0 ,0 ,1)]`
`=[(4, 2 ,3),( 1 ,1 ,2 ),(3 ,0 ,1)][(-2,2,-1),(3,0,3),(-1,4,-2)]`
`=[(7,20,-22),(-10,10,-1),(-5,10,5)]`
Clearly `(AB) C = A (BC) `
Promotional Banner

Topper's Solved these Questions

  • ADJOINTS AND INVERSE OF MATRIX

    RD SHARMA|Exercise QUESTION|1 Videos

  • ALGEBRA OF VECTORS

    RD SHARMA|Exercise Solved Examples And Exercises|331 Videos

Similar Questions

Explore conceptually related problems

For the following matrices verify the distributivity of matrix multiplication over matrix addition i.e.A(B+C)=AB+ACA=[1-102],B=[-1021] and C=[011-1]( ii) A=[2-111-12]B=[0111] and C=[1-101]

For the following matrices, verify that A(BC)=(AB)C : (i) A=[{:(1,2,5),(0,1,3):}],B=[{:(2," "3,0),(1," "0,4),(1,-1,2):}]" and "C=[{:(1),(4),(5):}] (ii) A=[{:(2,3,-1),(3,0," "2):}],B=[{:(1),(1),(2):}]" and "C=[1-2]

Verify that A(B+C)=(AB+AC), when (i) A=[{:(1,2),(3,4):}],B=[{:(2," "0),(1,-3):}]" and "C=[{:(1,-1),(0," "1):}]. (ii) A=[{:(2,3),(-1,4),(0,1):}],B=[{:(5,-3),(2," "1):}]" and "C=[{:(-1,2),(" "3,4):}].

Verify associative law of matrix additions for the matrices : A=[(1,0),(2,-1)],B=[(3,7),(4,8)] and C=[(-1,0),(0,0)] .

If A=[(1,2,3),(-1,0,2),(1,-8,-1)],B=[(4,5,6),(-1,0,1),(2,1,2)],C=[(-1,-2,1),(-1,2,3),(-1,-2,2)] , find (i) 2B-3C (ii) A-2B+3C .

If A=[{:(1,2),(-2,1):}],B=[{:(2,3),(3,-4):}] and C=[{:(1,0),(-1,0):}] , verfity (i) A(B+C)=AB+AC.

The inverse of the matrix [(1,0,0),(a,1,0),(b,c,1)] is (A) [(1,0,0),(-a,1,0),(b,c,1)] (B) [(1,0,0),(-a,1,0),(ac,b,1)] (C) [(1,-a,ac-b),(-0,1,-c),(0,0,1)] (D) [(1,0,0),(-a,1,0),(ac-b,-c,1)]

If A=[{:(" "1,-1," "2),(" "3," "2," "0),(-2," "0," "1):}],B=[{:(3,1),(0,2),(-2,5):}]" and "C=[{:(2,1,-3),(3,0,-1):} then verify that (AB)C=A(BC).

If A=[{:(3,2),(1,0):}],B=[{:(1,-2,5),(0,7,3):}]" and "C=[{:(8,1,-6),(2,-5,0):}] , verify that A(B+C)=(AB+AC).

Let A+B+C= [{:(,4,-1),(,0,1):}],4A+2B+C=[{:(,0,-1),(,-3,2):}]and 9A+3B+C=[{:(,0,2),(,2,1):}]"then find A"

RD SHARMA-ALGEBRA OF MATRICES-Solved Examples And Exercises
  1. If A=[(2,-3,-5),(-1, 4 ,5),( 1,-3,-4)] and B=[(2,-2,-4),(-1 ,3, 4 ),(1...

    Text Solution

    |

  2. Let A=[(-1, 1,-1 ),(3,-3, 3 ),(5, 5, 5)] and B=[(0 ,4 ,3),( 1,-3,-3),(...

    Text Solution

    |

  3. For the following matrices verify the associativity of matrix multip...

    Text Solution

    |

  4. For the following matrices verify the distributivity of matrix multi...

    Text Solution

    |

  5. If A=[(1, 0,-2 ),(3,-1, 0),(-2, 1, 1)] , B=[(0 ,5,-4),(-2, 1 ,3),(-1, ...

    Text Solution

    |

  6. Compute the elements a(43) and a(22) of the matrix: A=[(0 ,1, 0),( 2, ...

    Text Solution

    |

  7. If A=[(0 ,1 ,0),( 0, 0, 1),(p, q ,r)] and I is the identity matrix o...

    Text Solution

    |

  8. If omega is a complex cube root of unity, show that ([(1,omega,omega...

    Text Solution

    |

  9. If A=[(2,-3,-5),(-1 ,4 ,5),( 1,-3,-4)] , show that A^2=A .

    Text Solution

    |

  10. If A=[(4,-1,-4 ),(3, 0,-4),( 3,-1,-3)] , show that A^2=I3

    Text Solution

    |

  11. If [1\ \ 1\ \ x][1 0 2 0 2 1 2 1 0][1 1 1]=0 , find xdot

    Text Solution

    |

  12. If [(2,3),(5,7)][(1,-3),(-2,4)]=[(-4,6),(-9,x)] , find xdot

    Text Solution

    |

  13. If [(x,4,1)][(2,1,2),(1,0,2),(0,2,-4)][(x),(4),(-1)]=0 , find xdot

    Text Solution

    |

  14. If [(1, -1,x)][(0 ,1,-1),( 2 ,1 ,3),( 1 ,1 ,1)][(0),( 1),( 1)]=0 , fin...

    Text Solution

    |

  15. If A=[(3,-2),( 4,-2)] and I=[(1 ,0),( 0 ,1)] , then prove that A^2-A+2...

    Text Solution

    |

  16. If A=[(3 ,1),(-1, 2)] and I=[(1, 0), (0 ,1)] , then find lambda so tha...

    Text Solution

    |

  17. If A=[[3 ,1],[-1, 2]] , show that A^2-5A+7I2=O .

    Text Solution

    |

  18. If A=[(2, 3),(-1 ,0)] , show that A^2-2A+3I2=O .

    Text Solution

    |

  19. Show that the matrix A=[(2, 3), (1, 2)] satisfies the equation A^3-4A...

    Text Solution

    |

  20. Show that the matrix A=[(5, 3), (12 ,7)] is a root of the equation A...

    Text Solution

    |