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Matrices and Determinants Test - 2

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Matrices and Determinants Test - 2
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  • Question 1
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    Solution

    A square matrix A for which An = 0 , where n is a positive integer, is called a Nilpotent matrix.

  • Question 2
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    Solution






  • Question 3
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    If A any square matrix then which of the following is not symmetric ?

    Solution

    For every square matrix (A –A ’) is always skew –symmetric.

  • Question 4
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    Let a, b, c, d, u, v be integers. If the system of equations, a x + b y = u, c x + dy = v, has a unique solution in integers, then

    Solution


    ax + by = u , cx +dy = v , 

    since the solution is unique in integers.



  • Question 5
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    The system of equations, x + y + z = 1, 3 x + 6 y + z = 8, αx + 2 y + 3z = 1 has a unique solution for

    Solution

    The given system of equations has unique solution , if  


    ⇒1(18 −2)−1(9 −α) ⇒13 −5 α≠0 ⇒α≠13/5  + 1(6 −6 α) ≠0
    Therefore , unique solution exists for all integral values of  α.

  • Question 6
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    If A and B are symmetric matrices of the same order, then

    Solution

    If A and B are symmetric matrices of the same order, then , AB + BA is always a symmetric matrix.

  • Question 7
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    If  A = [aij ]2 ×2 where  aij = i  + j, then A is equal to

    Solution

    If A = [aij ]2x2  where aij  = i + j, then, 

  • Question 8
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    Each diagonal element of a skew-symmetric matrix is

    Solution

    The diagonal elements of a skew-symmetric is zero.

  • Question 9
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    The system of equations, x + y + z = 6, x + 2 y + 3 z = 14, x + 3 y + 5z = 20 has

    Solution

    The given system of equations does not has a solution if : 


     

    0 ⇒1(10 -9) - 1(5-3) + 1(3-2)

    = 0 ⇒1-2 + 1 = 0

  • Question 10
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    The matrix of the transformation ‘reflection in the line x + y = 0 ‘is

    Solution

    Let x 'and y 'be the reflection of x and y, therefore :



    Hence, reflection is on the line - x-y = 0 ⇒x + y = 0

  • Question 11
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    A square matrix  A = [aij ]n ×n  is called a diagonal matrix if  aij  = 0 for

    Solution

    In a diagonal matrix all elements except diagonal elements are zero.i.e. 

  • Question 12
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    If    and   then AB =

    Solution




  • Question 13
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    Solution

    or a symmetric matrix A ’= A . therefore , 

  • Question 14
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    If  In is the identity matrix of order n, the  (In)−1

    Solution

    Inverse of any identity matrix is always an identity matrix.

  • Question 15
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    If A = [x y z],    and C = [xyt]t , then ABC is

    Solution


    Matrix C is given in Transpose form. So, Matrix C is 3x1 order.

  • Question 16
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    If a square matrix A has two identical rows or columns , then det.A is :

    Solution

    Det.A = 0.

  • Question 17
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    For a skew symmetric odd ordered matrix A of integers, which of the following will hold true:

    Solution

    Determinant of a skew symmetric odd ordered matrix A is always 0 .

  • Question 18
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    Matrix A when multiplied with Matrix C gives the Identity matrix I, what is C?

    Solution

    Any square matrix when multiplied with its inverse gives the identity matrix. Note that non square matrices are not invertible.

  • Question 19
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    Let for any matrix M ,M−1 exist. Which of the following is not true.

    Solution

    Clearly , (M −1)−1  = (M−1 )1  is not true.

  • Question 20
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    Rank of a non-zero matrix is always

    Solution

    Rank of a non zero matrix is always greater than or equal to 1.

  • Question 21
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    For a non-trivial solution | A | is

    Solution

    An n ×n homogeneous system of linear equations has a unique solution (the trivial solution) if and only if its determinant is non-zero. If this determinant is zero, then the system has an infinite number of solutions. i.e. For a non-trivial solution ∣A ∣=0.

  • Question 22
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    If for a matrix A, A2 +I = O where I is the identity matrix, then A equals

    Solution



  • Question 23
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    Solution

    The given matrix is a skew –symmetric matrix.,therefore , A = - A ’.

  • Question 24
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    The system of linear equations x + y + z = 2, 2x + y - z = 3, 3x + 2y - kz = 4 has a unique solution if ,

    Solution

    The given system of equation has a unique solution if : 

  • Question 25
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    The value of k for which the system of equations, x + k y + 3 z = 0, 3 x + k y –2 z = 0, 2 x + 3 y –4 z = 0, have a non-trival solution is  

    Solution

    The given system of equations has a non-trivial solution if  :

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