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Question
A matrix which is both symmetric and skew symmetric matrix is a ______.
Options
triangular matrix
identity matrix
diagonal matrix
null matrix
Solution
A matrix which is both symmetric and skew symmetric matrix is a null matrix.
Explanation:
A matrix that is both symmetric and skew-symmetric must satisfy the properties of both types:
- Symmetric Matrix: A matrix A is symmetric if A = AT, meaning it is equal to its transpose.
- Skew-Symmetric Matrix: A matrix A is skew-symmetric if A = – AT, meaning it is equal to the negative of its transpose.
For a matrix to be both symmetric and skew-symmetric, we have:
A = AT and A = – AT
Combining these, we get:
A = – A
This implies that each element of the matrix must be zero:
Aij = – Aij
2Aij = 0
Aij = 0
Therefore, the only matrix that satisfies both conditions is the null matrix (a matrix where all elements are zero).
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