InterviewSolution
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InterviewSolution
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Which of the following is correct?68.(a)(b)(c)(d)Skew symmetric matrix of evenorder is always symmetricskew symmetric matrix of oddorder is non-singularskew symmetric matrix of oddorder is singularnone |
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Answer» Let, A be a skew-symmetric square matrix of n×n , where n is odd, By general properties of determinants, det(A)=det(AT)…(i)However, since A is a skew-symmetric matrix where aij=−aij (i,j are rows and column numbers ), ∴ In case of skew-symmetric matrix, AT=−ANowdet(−A)=det(AT)But,det(−A)=(−1)ndet(A)where n is no. of rows/columns in a square Matrix. ∴det(AT)=(−1)ndet(A)∵nisodd,(−1)n=−1∴det(AT)=−det(A)…(ii)Subtractingequation(ii)from(i),∴2det(A)=det(AT)−det(AT)=0∴det(A)=0Hence matrix A is singular.... By definition of singular matrix Hence proved. option (c) is correct i.e skew symmetric matrix of odd order is singular. not it's not crrct |
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