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Matrices and Determinants Test - 4

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Matrices and Determinants Test - 4
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  • Question 1
    2 / -0.83

    Solution



     

    ⇒(x-y)(y-z)[y2  + yz+z2  - x2  - xy - y2 ]

    ⇒(x-y)(y-z) (z-x) (x+y+z)

  • Question 2
    2 / -0.83

    If    , then   is equal to  

    Solution


    =1(-5) -2(10) + 3(11)
    =-5-20+33 = 8

    =1(-30) - 6(20) + 3(66)

    = -30-120+198= 48

    D = 8 ⇒6D = 48

  • Question 3
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    Solution

    Apply, R2  →R2  - R1 ,


    Apply, R3  →R3  - 4R1 ,

     

    ⇒(x-3) (6x -9)  = 0 ⇒x = 

  • Question 4
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    If the order of matrix A is m*p. And the order of B is p ×n. Then the order of matrix AB is ?

  • Question 5
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    If each element of a  3  × 3  matrix A is multiplied by 3 , then the determinant of the newly formed matrix is

  • Question 6
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    One root of the equation   

    Solution


    Apply , R1 →R1 +R2 +R3 ,
     


    ⇒either (3x -2) = 0

  • Question 7
    2 / -0.83

    Solution

    Apply , C1  →C1 +C2 +C3 +C4


    Apply, R1 →R1  - R2 ,


    Apply, R1 →R1  - R2

    ⇒ (10+ x) x3

  • Question 8
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    The determinant   is equal to  

    Solution

    Apply, C1  →C1   + C2  + C3 ,

  • Question 9
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    Solution

    The determinant of a lower triangular (or an upper triangular matrix is equal to the product of the diagonal elements.

  • Question 10
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    If A is a non singular matrix of order 3 , then  |adj(adjA)|

    Solution

    where n is order of matrix. Here n = 3.

  • Question 11
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    Solution



  • Question 12
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    The only integral root of the equation det.   is

    Solution

     

    Clearly , y = 1 satisfies it. [C3 = 3C1 ]

  • Question 13
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    If 1/a+1/b+ 1/c = 0 , then   

    Solution

    Since , 

  • Question 14
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    If A is a non singular matrix and A ’denotes the transpose of A , then

    Solution

    Because , |A|=|A ′|

  • Question 15
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    If the matrix AB = O , then

    Solution

    If the matrix AB = O , then , marix A can be a non zero matrix as well as matrix B can be a non zero matrix. Which means det.A = 0 and det.B = 0.

  • Question 16
    2 / -0.83

    If A ,B andC be the three square matrices such that A = B + C , then Det A is equal to

    Solution

    Because , |A| ≠|B|+|C|

  • Question 17
    2 / -0.83

    Solution set of the equation  

    Solution

    [x(-3x(x+2) - 2x(x-3)]+6[2(x+2)+3(x-3)]-1(4x-9x) = 0

    ⇒- 5x3  + 35x - 30 = 0
    ⇒(x-1)(x-2)(x+3) = 0 ⇒x=1,2,-3

  • Question 18
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    A determinant is unaltered , if

    Solution

    This is because of the elementary transformations of determinants . The value of determinant remains unaffected by applying elementary transformations.

  • Question 19
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    If A and B are square matrices of order 3 , such that Det.A = –1 , Det.B = 3 then the determinant of 3AB is equal to

    Solution

     

    |3AB| = 27|A||B| = 27(-1)(3) = -81

  • Question 20
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    Solution

    Because , the determinant of a skew symmetric matrix of odd order is always zero and of even order is a non zero perfect square.

  • Question 21
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    If 1 , ω,ω2 are cube roots of unity , then    has value

    Solution

     

    write  1  as  in  R1  and  take  out   common  from  R1  , we  get  : 


    because row 1 and row 3 are identical.

  • Question 22
    2 / -0.83

    Solution set of the equation  

    Solution

    x(x2  - 12) - 3(2x - 14)+7(12 - 7x) = 0

    ⇒x3  - 67x + 126 = 0

    ⇒(x-2)(x-7)(x+9) = 0 ⇒x = 2,7,-9

  • Question 23
    2 / -0.83

      

    Solution



  • Question 24
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    The system AX = B of n equations in n unknowns has infinitely many solutions if

    Solution

    Explanation here if det. A = 0 , (adj A) B = O ⇒ The system AX = B of n equations in n unknowns may be consistent with infinitely many solutions or it may be inconsistent.

  • Question 25
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    If A is a square matrix such that  A3  = I , then  A−1  is equal to

    Solution

    A3  = I ⇒ Pre - multiplying both sides by  A−1 ,A−1 , A3  = A−1  I ⇒A2  = A−1

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