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If A is a skew – symmetric matrix and n is an even positive integer, then An is
- a)A symmetric matrix
- b)A skew–symmetric matrix
- c)A diagonal matrix
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
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If A is a skew – symmetric matrix and n is an evenpositive integ...
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Skew-symmetric Matrix and its Properties:
A skew-symmetric matrix is a square matrix whose transpose is equal to its negative. In other words, a matrix A is skew-symmetric if A^T = -A.
Properties of Skew-symmetric Matrices:
- The main diagonal of a skew-symmetric matrix consists of zeros.
- If A is a skew-symmetric matrix, then A^n (where n is a positive even integer) is a symmetric matrix.
Understanding the Given Statement:
Given that A is a skew-symmetric matrix and n is an even positive integer, we need to determine the nature of the matrix A^n.
Analysis:
- Since A is skew-symmetric, A^T = -A.
- When we raise a skew-symmetric matrix to a power, the resulting matrix may not necessarily be skew-symmetric. However, when n is an even number, A^n will be a symmetric matrix.
Conclusion:
Therefore, if A is a skew-symmetric matrix and n is an even positive integer, the resulting matrix A^n will be a symmetric matrix. Thus, the correct answer is option 'A' - a symmetric matrix.
A skew-symmetric matrix is a square matrix whose transpose is equal to its negative. In other words, a matrix A is skew-symmetric if A^T = -A.
Properties of Skew-symmetric Matrices:
- The main diagonal of a skew-symmetric matrix consists of zeros.
- If A is a skew-symmetric matrix, then A^n (where n is a positive even integer) is a symmetric matrix.
Understanding the Given Statement:
Given that A is a skew-symmetric matrix and n is an even positive integer, we need to determine the nature of the matrix A^n.
Analysis:
- Since A is skew-symmetric, A^T = -A.
- When we raise a skew-symmetric matrix to a power, the resulting matrix may not necessarily be skew-symmetric. However, when n is an even number, A^n will be a symmetric matrix.
Conclusion:
Therefore, if A is a skew-symmetric matrix and n is an even positive integer, the resulting matrix A^n will be a symmetric matrix. Thus, the correct answer is option 'A' - a symmetric matrix.
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If A is a skew – symmetric matrix and n is an evenpositive integer, then Anisa)Asymmetric matrixb)A skew–symmetric matrixc)A diagonal matrixd)None of theseCorrect answer is option 'A'. Can you explain this answer?
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