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Adjoint of a square matrix

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Adjoint of square matrices and their properties

Let B be a skew symmetric matrix of order 3times3 with real entries. Given I-B and I+B are non-singular matrices. If A=(I+B)(I-B)^(-1), where det (A)>0 ,then find the value of det(2A)-det(adj(A)) [Note: det(P) denotes determinant of square matrix P and det(adj (P)) denotes determinant of adjoint of square matrix P respectively.]

Knowledge Check

  • If the matrix B is the adjoint of the square matrix A and alpha is the value of the determinant of A, then what is AB equal to ?

    A
    `alpha`
    B
    `((1)/(alpha))I`
    C
    I
    D
    `alpha I`

  • Let matrix A=[(x,y,-z),(1,2,3),(1,1,2)] where x,y, z in N . If det. (adj. (adj. A)) =2^(8)*3^(4) then the number of such matrices A is : [Note : adj. A denotes adjoint of square matrix A.]

    A
    220
    B
    45
    C
    55
    D
    110

  • If D is the determinant of a square matrix A of order n, then the determinant of its adjoint is

    A
    `D`
    B
    `D^(n-1)`
    C
    `D^(n)`
    D
    `D^(n+1)`

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    If d is the determinant of a square matrix A of order n , then the determinant of its adjoint is d^n (b) d^(n-1) (c) d^(n+1) (d) d

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    Which of the following is/are incorrect? (i) adjoint of a symmetric matrix is symmetric (ii) adjoint of a unit matrix is a unit matrix (iii) A (adj A)=(adj A)A=absAI (iv) adjoint of a diagonal matrix is a diagonal matrix

    Which of the following is/are incorrect? (i).Adjoint of symmetric matrix is symmetric (ii).Adjoint of unit matrix is a unit matrix (iii). A(adjA)=(adjA)A=!|A|I (iv). Adjoint of a diagonal matrix is a diagonal matrix

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