Home
Class 12
MATHS
LetA= [[1,0,0],[2,1,0],[3,2,1]] be a squ...

Let`A= [[1,0,0],[2,1,0],[3,2,1]]` be a square matrix and ` C_(1), C_(2), C_(3)` be three
comumn matrices satisgying `AC_(1) = [[1],[0],[0]], AC_(2) = [[2],[3],[0]] and AC_(3)= [[2],[3],[1]]` of matrix B. If the matrix `C= 1/3 (AcdotB).`
The ratio of the trace of the matrix B to the matrix C, is

A

`-9/5`

B

`-5/9`

C

`-2/3`

D

`-3/2`

Text Solution

Verified by Experts

The correct Answer is:
A
Let ` B = [[alpha_(1),beta_(1) ,gamma_(1)],[alpha_(2),beta_(2) ,gamma_(2)],[alpha_(3),beta_(3) ,gamma_(3)]]`
`therefore C_(1)=[[alpha_(1)],[alpha_(2)],[alpha_(3)]], C_(2) = [[beta_(1)],[beta_(2)],[beta_(3)]]and C_(3) = [[gamma_(1)],[gamma_(2)],[gamma_(3)]]`
`rArr AC_(1) = [[alpha_(1)],[alpha_(1)+alpha_(2)],[3alpha_(1)+alpha_(2)+alpha_(3)]] = [[1],[0],[0]]`
`rArr alpha _(1) = 1 , alpha _(2) = -2, alpha _(3) = 1`
`rArr AC_(2) = [[beta_(1)],[2beta_(1)+beta_(2)],[3beta_(1)+2beta(2)+beta_(3)]] = [[2],[3],[0]]`
`beta _(1) = 2, beta_(2) = -1, beta_(3) = -4`
and `AC_(3) = [[gamma_(1)],[2gamma_(1)+gamma_(2)],[3gamma_(1)+2gamma_(2)+gamma_(3)]] = [[2],[3],[1]]`
`rArr gamma_(1)= 2, gamma_(2) = -1, gamma_(3) = -3`
`therefore B = [[1,2,2],[-2,-1,-1],[1,-4,-3]]`
`rArr det B = [[1,2,2],[-2,-1,-1],[1,-4,-3]]`
` = 1 (3-4) - 2 (6+1) + 2(8+1) = 3`
and `C=1/3 [[1,0,0],[2,1,0],[3,2,1]] [[1, 2, 2 ],[-2,-1, -1],[1, -4, -3]]`
`= 1/ 3 [[1,2,2],[0,3,3],[0,0,1]]`
`therefore det C= abs[[1/3,2/3,2/3],[0,1,1],[0,0,1/3]] = 1/9`
`("trace of B")/("trace C")= ((-3))/((5/3))=-9/5`
Promotional Banner

Topper's Solved these Questions

  • MATRICES

    ARIHANT MATHS|Exercise Exercise (Single Integer Answer Type Questions)|10 Videos

  • MATRICES

    ARIHANT MATHS|Exercise Matrices Exercise 5 : (Matching Type Questions )|4 Videos

  • MATRICES

    ARIHANT MATHS|Exercise Exercise (More Than One Correct Option Type Questions)|15 Videos

  • MATHEMATICAL INDUCTION

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|2 Videos

  • MONOTONICITY MAXIMA AND MINIMA

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|31 Videos

Similar Questions

Explore conceptually related problems

Let A= [[1,0,0],[2,1,0],[3,2,1]] be a square matrix and C_(1), C_(2), C_(3) be three comumn matrices satisgying AC_(1) = [[1],[0],[0]], AC_(2) = [[2],[3],[0]] and AC_(3)= [[2],[3],[1]] of matrix B. If the matrix C= 1/3 (AcdotB). The value of det(B^(-1)) , is

Let A= [[1,0,0],[2,1,0],[3,2,1]] be a square matrix and C_(1), C_(2), C_(3) be three comumn matrices satisgying AC_(1) = [[1],[0],[0]], AC_(2) = [[2],[3],[0]] and AC_(3)= [[2],[3],[1]] of matrix B. If the matrix C= 1/3 (AcdotB). find sin^(-1) (detA) + tan^(-1) (9det C)

Let A=[(1,0,0),(2,1,0),(3,2,1)], it U_1, U_2 and U_3 are column matrices satisfying AU_1 =[(1),(0),(0)], AU_2=[(2),(3),(0)] and AU_3=[(2),(3),(1)] and U is a 3xx3 matrix when columns are U_1,U_2,U_3 now answer the following question: The value of |U| is (A) 3 (B) -3 (C) 3/2 (D) 2

Let A=[(-1, 2, -3),(-2,0,3),(3,-3, 1)] be a matrix, then |a| adj(A^(-1)) is equal to

Let A=[(1,0,0),(2,1,0),(3,2,1)], if U_1, U_2 and U_3 are column matrices satisfying AU_1 =[(1),(0),(0)], AU_2=[(2),(3),(0)] and AU_3=[(2),(3),(1)] and U is a 3xx3 matrix when columns are U_1,U_2,U_3 now answer the following question: The sum of elements of U^-1 is: (A) -1 (B) 0 (C) 1 (D) 3

If A=[[1,2,-3] , [5,0,2] , [1,-1,1]] and B=[[3,-1,2] , [4,2,5] , [2,0,3]] then find matrix C such that A+2C=B

Given A=[[1,2,-3],[5,0,2],[1,-1,1]] and B=[[3,-1,2],[4,2,5],[2,0,3]] , find the matrix C such that A+C=B.

If A=[[2,3,4],[-3,0,2]], B=[[3,-4,-5],[1,2,1]] and C=[[5,-1,2],[7,0,3]] , find the matrix X such that 2A+3B=X+C

In the matrix [[1,0,5],[2,-3,4]] order of matrix matrix is………

If A=[[1,3,2],[-1,2,0],[0,-1,1]] , then find a matrix X such that A^(2)-2A-X=0

ARIHANT MATHS-MATRICES -Exercise (Passage Based Questions)
  1. Suppose A and B be two ono-singular matrices such that AB= BA^(m), B...

    Text Solution

    |

  2. Suppose A and B be two ono-singular matrices such that AB= BA^(m), B...

    Text Solution

    |

  3. Suppose A and B be two ono-singular matrices such that AB= BA^(m), B...

    Text Solution

    |

  4. Let A= [[a,b,c],[b,c,a],[c,a,b]] is an orthogonal matrix and abc = lam...

    Text Solution

    |

  5. Let A= [[a,b,c],[b,c,a],[c,a,b]] is an orthogonal matrix and abc = lam...

    Text Solution

    |

  6. Let A= [[a,b,c],[b,c,a],[c,a,b]] is an orthogonal matrix and abc = lam...

    Text Solution

    |

  7. LatA = [a(ij)](3xx 3). If tr is arithmetic mean of elements of rth row...

    Text Solution

    |

  8. LatA = [a(ij)](3xx 3). If tr is arithmetic mean of elements of rth row...

    Text Solution

    |

  9. LetA= [[1,0,0],[2,1,0],[3,2,1]] be a square matrix and C(1), C(2), C(...

    Text Solution

    |

  10. LetA= [[1,0,0],[2,1,0],[3,2,1]] be a square matrix and C(1), C(2), C(...

    Text Solution

    |

  11. LetA= [[1,0,0],[2,1,0],[3,2,1]] be a square matrix and C(1), C(2), C(...

    Text Solution

    |

  12. If A is symmetric and B skew- symmetric matrix and A + B is non-sing...

    Text Solution

    |

  13. If A is a symmetric and B skew symmetric matrix and (A+ B) is non-sing...

    Text Solution

    |

  14. If A is symmetric and B skew- symmetric matrix and A + B is non-sin...

    Text Solution

    |

  15. Let A be a squarre matrix of order of order 3 satisfies the matrix equ...

    Text Solution

    |

  16. Let A be a squarre matrix of order of order 3 satisfies the matrix equ...

    Text Solution

    |