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    Question

    Let A be a 3×3 matrix such that AT =−A and

    all diagonal entries are zero. Which of the following is always true ?
    A A² = I Correct Answer Incorrect Answer
    B A is symmetric Correct Answer Incorrect Answer
    C All eigenvalues of A are purely imaginary or zero Correct Answer Incorrect Answer
    D A is diagonalizable over real numbers Correct Answer Incorrect Answer

    Solution

    We are given:

    • A is a 3×3 matrix
    • Aᵀ = –A, so A is skew-symmetric
    • All diagonal entries are zero (which is always true for real skew-symmetric matrices)
    We are to determine which of the following is always true . All eigenvalues of A are purely imaginary or zero This is always true for real skew-symmetric matrices. The eigenvalues of such matrices are either:
    • Zero, or
    • Purely imaginary complex numbers, which come in conjugate pairs.
    Since the matrix is 3×3, if it has non-real eigenvalues, two of them must be complex conjugates, and the third must be real. The only real number possible in this case is zero. So the eigenvalues must be 0, iλ, –iλ where λ is real. This statement is correct. A is symmetric This contradicts the given condition Aᵀ = –A. A symmetric matrix satisfies Aᵀ = A, which is the opposite of skew-symmetric. This statement is incorrect. A² = I For A² = I to hold, A must be an involutory matrix. That is not generally true for skew-symmetric matrices. In fact, for skew-symmetric matrices, A² is usually a negative semi-definite matrix, not the identity. This statement is incorrect. A is diagonalizable over real numbers Real skew-symmetric matrices may have complex eigenvalues, so they are not necessarily diagonalizable over the real numbers. They are diagonalizable over complex numbers, but the question specifically asks about real numbers. This statement is incorrect.

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